{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-06T14:01:22.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-04-06T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":52467,"title":"Easy Sequences 2: Trigonometric function with integral input and output","description":"The function 'F', defined as: \r\n\r\n                ,\r\n\r\nwill always return an integer if the input is a natural number (in radians). Furthermore, since the cosine and sine of 0, are 1 and 0 respectively, therefore F(0) = 0.  For a given natural number 'n'  your task is to find the value of F(n).","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440000534057617px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 163px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eThe function 'F', defined as: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"403.5\" height=\"21\" style=\"width: 403.5px; height: 21px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003ewill always return an integer if the input is a natural number (in radians). Furthermore, since the cosine and sine of 0, are 1 and 0 respectively, therefore F(0) = 0.  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"font-weight: bold; \"\u003eFor a given natural number 'n'  your task is to find the value of F(n).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = F(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 0;\r\nassert(isequal(F(x),0))\r\n%%\r\nx = 10;\r\nassert(isequal(F(x),round(-cos(2*pi*x/3)/9+(2*x*sqrt(3)/9+1/sqrt(3))*sin(2*pi*x/3)+1/9)))\r\n%%\r\nx = 20;\r\nassert(isequal(F(x),round(-cos(2*pi*x/3)/9+(2*x*sqrt(3)/9+1/sqrt(3))*sin(2*pi*x/3)+1/9)))\r\n%%\r\nx = intmax-4\r\nassert(isequal(F(x),F(1234567891011)))\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":43,"test_suite_updated_at":"2021-08-11T12:47:13.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-10T11:56:54.000Z","updated_at":"2026-04-01T20:57:52.000Z","published_at":"2021-08-11T07:18:18.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe function 'F', defined as: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF(x) = -\\\\cos(2\\\\pi x/3)/9+(2x\\\\sqrt{3}/9+1/\\\\sqrt{3})\\\\cdot\\\\sin(2\\\\pi x/3)+1/9\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewill always return an integer if the input is a natural number (in radians). Furthermore, since the cosine and sine of 0, are 1 and 0 respectively, therefore F(0) = 0.  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFor a given natural number 'n'  your task is to find the value of F(n).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52462,"title":"Easy Sequences 1: Find the index of an element","description":"The nth element of a series  is defined by: . Obviously, the first element . Given the nth element , find the value of the corresponding index .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 66px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 33px; transform-origin: 407px 33px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 33px; text-align: left; transform-origin: 384px 33px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 88.5px 8px; transform-origin: 88.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe nth element of a series \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eA\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 46px 8px; transform-origin: 46px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is defined by: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 155.5px; height: 45px;\" width=\"155.5\" height=\"45\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 91px 8px; transform-origin: 91px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Obviously, the first element \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 59px; height: 18.5px;\" width=\"59\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 36.5px 8px; transform-origin: 36.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Given the nth element \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEIAAAAlCAYAAAD2pT8KAAACZklEQVRoge2ZYZGDMBCFPw84wEANoKAKcFAHODgLaEACHmqhGmqB+wFv2NKUbHoJcz94M525uZZs9mX3vSTAiRMnTvx/3IAq01g1cM001qHogTbzmN0y7uFo+G5Fe+ZqKIGeg8logYmZjBR0wJh/Oi8YyV9tQVTAg5mIlIDN8kxdYlIGF+B5QBw65oQm4CfhuZHjyrancOXVrCRMwN353FHVsI13KRVgZG2LafnbI5gDftJy4UGhCrSraqsiRkRFehvlQM+sFdnxYE3mzkpEbCNzdf5ui4r30q6ZF8TTYjcKtEfHzK5Wf2AlIrYnkLjGrPbKaq/Pzdj18n/FfBJPUBWczUqrJbBNOMU5RFoMWm2rQTWrHXaJcUVE54jtQs+70KncPc6hlfRCbTcyE2FXX8l5V3oik2Aq8La/L2ZCMedIIULCqra485rwzXzvcSsR+meMy2SawMfrHClE2EoLbcB60pLLQoTOE+OHjwQt5ggpRChRVdoWiunp+yy2LYHcG8TrHErOAyuUe+3oscQsYil13it5r4J77dMmOgS+lz54N0mf9M0N7RxjJaXWiTmH18+tEIYmrxbzuoAW4OvzjUo+pspWMGPOEWszWBMNrbh1ExE6sN8iA2GdcUHJeVi3pRxjvndMSuOEYtvqq5bfxIj9Wh+0g/MSYa0uFlQEf9IJO1aoLWzb3AlriIWIS26Lhnd77D4M1LCeCbzPwP7FTGvGCKFiTt57DTdy/GnXDW2XU+85U3Hl9YD4L9Eyl3apSWr/U5rsLCj5/uGwG+xcaMlPRomXRofg2xdEIeje4sSJEydO5MYvnSP5KzErv3UAAAAASUVORK5CYII=\" style=\"width: 33px; height: 18.5px;\" width=\"33\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 134.5px 8px; transform-origin: 134.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, find the value of the corresponding index \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function n = index(a)\r\n  n = a;\r\nend","test_suite":"%%\r\na = 1;\r\nn = index(a);\r\nassert(isequal(1,n))\r\n%%\r\na = 25;\r\nn = index(a);\r\nbs = arrayfun(@(x) sum(arrayfun(@(k) k*(-1)^(k^3+1),1:x)),1:n);\r\nb = bs(end);\r\nassert(isequal(a,b))\r\n%%\r\na = 100;\r\nn = index(a);\r\nbs = arrayfun(@(x) sum(arrayfun(@(k) k*(-1)^(k^3+1),1:x)),1:n);\r\nb = bs(end);\r\nassert(isequal(a,b))\r\n%%\r\na = randi([1000,ceil(exp(log(double(intmax)/2)))]);\r\nn = index(a);\r\nassert(isequal(index(-a+(1-(-1)^(n+1))/2),n+1))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":9,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":52,"test_suite_updated_at":"2021-08-11T04:47:44.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-10T10:31:40.000Z","updated_at":"2026-04-01T20:40:04.000Z","published_at":"2021-08-10T10:34:24.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe nth element of a series \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eA\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is defined by: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eA(n) = \\\\displaystyle\\\\sum\\\\limits_{k=1}^n (k\\\\cdot(-1)^{k^3+1})\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Obviously, the first element \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eA(1) = 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Given the nth element \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eA(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, find the value of the corresponding index \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52594,"title":"Easy Sequences 10: Sum of Cumsums of Fibonacci Sequence","description":"The function F(n) is defined as the set of Fibonacci numbers from the first up to the n-th. S(n) is the result of applying to F, the MATLAB built-in function 'cumsum' and followed by the function 'sum'. For example for n = 10, we have:\r\n  \u003e\u003e Fn = [1 1 2 3 5 8 13 21 34 55];\r\n  \u003e\u003e Sn = sum(cumsum(Fn))\r\n  \u003e\u003e Sn =\r\n     364\r\nIt has been argued that the Fibonacci sequence exhibits an exponentially growth relative to the golden ratio, which means that S(n) becomes very big, very quickly. So instead, we will do the inverse operation. \r\nGiven a number 's', our task is to find the largest 'n', such that S(n) is less than or equal to 's'. In the example above we have, 'n = 10' for 's = 364' . ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 236.733px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 118.367px; transform-origin: 407px 118.367px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 130.5px 8px; transform-origin: 130.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe function F(n) is defined as the set of \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Fibonacci_number\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eFibonacci numbers\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 188.5px 8px; transform-origin: 188.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e from the first up to the n-th. S(n) is the result of applying to F, the MATLAB built-in function 'cumsum' and followed by the function 'sum'. For example for n = 10, we have:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 81.7333px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 40.8667px; transform-origin: 404px 40.8667px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 144px 8.5px; tab-size: 4; transform-origin: 144px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt; Fn = [1 1 2 3 5 8 13 21 34 55];\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 100px 8.5px; tab-size: 4; transform-origin: 100px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt; Sn = sum(cumsum(Fn))\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 36px 8.5px; tab-size: 4; transform-origin: 36px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt; Sn =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 32px 8.5px; tab-size: 4; transform-origin: 32px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     364\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 383.5px 8px; transform-origin: 383.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIt has been argued that the Fibonacci sequence exhibits an exponentially growth relative to the golden ratio, which means that S(n) becomes very big, very quickly. So instead, we will do the inverse operation.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 327px 8px; transform-origin: 327px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eGiven a number 's', our task is to find the largest 'n', such that S(n) is less than or equal to 's'. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.5px 8px; transform-origin: 47.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the example above we have,\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68px 8px; transform-origin: 68px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e'n = 10' for 's = 364' . \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function n = N(s)\r\n    n = inv_cumsum(inv_sum(s));\r\nend","test_suite":"%%\r\ns = 364;\r\nn_correct = 10;\r\nassert(isequal(N(s),n_correct))\r\n%%\r\ns = 2000;\r\nn_correct = 13;\r\nassert(isequal(N(s),n_correct))\r\n%%\r\ns = 5000:100:10000;\r\nn_correct = 798;\r\nn_answer = sum(arrayfun(@(i) N(i),s));\r\nassert(isequal(n_answer,n_correct))\r\n%%\r\ns = intmax;\r\nn_correct = 42;\r\nassert(isequal(N(s),n_correct))\r\n%%\r\ns = 10^10;\r\nn_correct = 45;\r\nassert(isequal(N(s),n_correct))\r\n%%\r\ns = intmax('uint64');\r\nn_correct = 89;\r\nassert(isequal(N(s),n_correct))\r\n%%\r\ns = realmax/10;\r\nn_correct = 1467;\r\nassert(isequal(N(s),n_correct))","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":49,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-23T07:31:32.000Z","updated_at":"2025-12-16T04:43:30.000Z","published_at":"2021-08-23T12:03:43.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe function F(n) is defined as the set of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Fibonacci_number\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eFibonacci numbers\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e from the first up to the n-th. S(n) is the result of applying to F, the MATLAB built-in function 'cumsum' and followed by the function 'sum'. For example for n = 10, we have:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[  \u003e\u003e Fn = [1 1 2 3 5 8 13 21 34 55];\\n  \u003e\u003e Sn = sum(cumsum(Fn))\\n  \u003e\u003e Sn =\\n     364]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt has been argued that the Fibonacci sequence exhibits an exponentially growth relative to the golden ratio, which means that S(n) becomes very big, very quickly. So instead, we will do the inverse operation.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven a number 's', our task is to find the largest 'n', such that S(n) is less than or equal to 's'. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eIn the example above we have,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e'n = 10' for 's = 364' . \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52584,"title":"Easy Sequences 9: Faithful Pairs","description":"A \"faithful number\" is a non-prime number that is one less or one more than some prime number but not both. For example, for numbers up to 20, the numbers 1 8, 10, 14, 16 and 20 are faithful. The number 4 does not qualify because it is equal to \"3 + 1\" and \"5 - 1\". \r\nIf both 'x' and 'x+2' are faithful but not to the same prime, the pair (x, x+2) is called a faithful pair. So, from 1 to 20 the faithful pairs are (8, 10) and (14, 16). Faithful pairs are scarse and rarer than primes themselves. We can only find 1 faithful pair for numbers 1 to 10, 5 pairs for numbers up to 50 and 8 pairs up to 100.\r\nLet \"P\" be the set of all faithful pairs from 1 to a given number \"n\". We define \"F\" as the set of all p1, p1 \u003c p2 ∀pairs (p1,p2) ∈ P. Write a function \"S(n)\", that sums all the elements of F. \r\nFor 1 to 20, P(20) = [8 10; 14 16], F(20) = [8 14] and S(20) = 22. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440000534057617px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 237px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eA \"faithful number\" is a non-prime number that is one less or one more than some prime number but not both. For example, for numbers up to 20, the numbers 1 8, 10, 14, 16 and 20 are faithful. The number 4 does not qualify because it is equal to \"3 + 1\" and \"5 - 1\". \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eIf both 'x' and 'x+2' are faithful but not to the same prime, the pair (x, x+2) is called a faithful pair. So, from 1 to 20 the faithful pairs are (8, 10) and (14, 16). Faithful pairs are scarse and rarer than primes themselves. We can only find 1 faithful pair for numbers 1 to 10, 5 pairs for numbers up to 50 and 8 pairs up to 100.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eLet \"P\" be the set of all faithful pairs from 1 to a given number \"n\". We define \"F\" as the set of all p1, p1 \u0026lt; p2 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e∀\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003epairs (p1,p2) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e∈\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e P. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"font-weight: bold; \"\u003eWrite a function \"S(n)\", that sums all the elements of F.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eFor 1 to 20, P(20) = [8 10; 14 16], F(20) = [8 14] and S(20) = 22. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function s = S(n)\r\n    n = 20;\r\n    p = [8 10; 14 16];\r\n    f = [8 14];\r\n    s = 22;\r\nend\r\n","test_suite":"%%\r\nn = 20;\r\ns_correct = 22;\r\nassert(isequal(S(n),s_correct))\r\n%%\r\nn = 9;\r\ns_correct = 0;\r\nassert(isequal(S(n),s_correct))\r\n%%\r\nn = 5:5:100;\r\ns_correct = [0 8 8 22 42 42 42 80 80 124 124 124 124 192 192 192 272 272 272 370];\r\nassert(isequal(S(n),s_correct))\r\n%%\r\nn = 1000;\r\ns_correct = 17216;\r\nassert(isequal(S(n),s_correct))\r\n%%\r\nn = 2^20;\r\ns_correct = 4054100250;\r\nassert(isequal(S(n),s_correct))\r\n%%\r\nn = intmax;\r\ns_correct = 6921757389660954;\r\nassert(isequal(S(n),s_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":19,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-22T10:45:18.000Z","updated_at":"2025-11-30T19:31:23.000Z","published_at":"2021-08-22T11:03:11.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA \\\"faithful number\\\" is a non-prime number that is one less or one more than some prime number but not both. For example, for numbers up to 20, the numbers 1 8, 10, 14, 16 and 20 are faithful. The number 4 does not qualify because it is equal to \\\"3 + 1\\\" and \\\"5 - 1\\\". \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf both 'x' and 'x+2' are faithful but not to the same prime, the pair (x, x+2) is called a faithful pair. So, from 1 to 20 the faithful pairs are (8, 10) and (14, 16). Faithful pairs are scarse and rarer than primes themselves. We can only find 1 faithful pair for numbers 1 to 10, 5 pairs for numbers up to 50 and 8 pairs up to 100.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eLet \\\"P\\\" be the set of all faithful pairs from 1 to a given number \\\"n\\\". We define \\\"F\\\" as the set of all p1, p1 \u0026lt; p2 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e∀\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003epairs (p1,p2) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e∈\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e P. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eWrite a function \\\"S(n)\\\", that sums all the elements of F.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor 1 to 20, P(20) = [8 10; 14 16], F(20) = [8 14] and S(20) = 22. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52497,"title":"Easy Sequences 3: Prime 44-number Squares","description":"The positive integers 62 and 238 are related. Their squares (3844 and 56,644) both end in '44'. In fact, 62 and 238 are the 3rd and 10th positive integers, respectively, that have this property. We will call a positive number whose square ends in '44' as a \"44-number\".\r\nIf 'x' is the nth 44-number, we define the function 'S(n)' to be the sum of the digits of 'x^2' but excluding the ending '44'. So in the cases above, S(3) = 11 and S(10) = 17. We noticed that both of these sums are primes.We define 'P(n)' as the number of prime S(n)'s among the first 'n' 44-numbers. Write a function that returns P(n), given that P(3) = 2 and P(10) = 5.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 156px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 78px; transform-origin: 407px 78px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 377.5px 8px; transform-origin: 377.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe positive integers 62 and 238 are related. Their squares (3844 and 56,644) both end in '44'. In fact, 62 and 238 are the 3rd and 10th positive integers, respectively, that have this property. We will call a positive number whose square ends in '44' as a \"44-number\".\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 382.5px 8px; transform-origin: 382.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf 'x' is the nth 44-number, we define the function 'S(n)' to be the sum of the digits of 'x^2' but excluding the ending '44'. So in the cases above, S(3) = 11 and S(10) = 17. We noticed that both of these sums are primes.We define 'P(n)' as the number of prime S(n)'s among the first 'n' 44-numbers. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 117px 8px; transform-origin: 117px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eWrite a function that returns P(n),\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 76.5px 8px; transform-origin: 76.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e given that P(3) = 2 and P(10) = 5.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function prime_count = P(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = [1 3 10 15 20];\r\ny_correct = [0 2 5 8 11];\r\nassert(isequal(arrayfun(@(i) P(i),x),y_correct))\r\n%%\r\nx = 1:20;\r\ny_correct = 108;\r\nassert(isequal(sum(arrayfun(@(i) P(i),x)),y_correct))\r\n%%\r\nx = arrayfun(@(i) P(i),15:30);\r\ny_correct = 118;\r\nassert(isequal(sum(arrayfun(@(i) P(i),x)),y_correct))\r\n%%\r\nx = 25:100;\r\ny_correct = 3077;\r\nassert(isequal(sum(arrayfun(@(i) P(i),x)),y_correct))\r\n%%\r\nx = floor(sqrt(double(intmax)));\r\ny_correct = 17862;\r\nassert(isequal(P(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":32,"test_suite_updated_at":"2021-08-12T04:00:36.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2021-08-11T10:45:05.000Z","updated_at":"2025-11-30T19:35:26.000Z","published_at":"2021-08-11T19:07:08.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe positive integers 62 and 238 are related. Their squares (3844 and 56,644) both end in '44'. In fact, 62 and 238 are the 3rd and 10th positive integers, respectively, that have this property. We will call a positive number whose square ends in '44' as a \\\"44-number\\\".\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf 'x' is the nth 44-number, we define the function 'S(n)' to be the sum of the digits of 'x^2' but excluding the ending '44'. So in the cases above, S(3) = 11 and S(10) = 17. We noticed that both of these sums are primes.We define 'P(n)' as the number of prime S(n)'s among the first 'n' 44-numbers. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eWrite a function that returns P(n),\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e given that P(3) = 2 and P(10) = 5.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52522,"title":"Easy Sequences 4: Eliminate the Days of Confusion","description":"If a date is written in purely numeric format some dates can be confusing, if we don't know the date format used. For example, \"05-02-1998\", may mean either \"May 02, 1998\" or \"February 05, 1998\". However, since there are only 12 months in a year, not all dates are confusing. \"23-10-1969\" is clearly \"October 23, 1969\".\r\nWrite a program that will count the number of days between two dates (inclusive) that are NOT confusing on either \"dd-mm-yyyy\" or \"mm-dd-yyyy\" formats.\r\nTo avoid further confusion you are given the input in \"yyyy-mm-dd\" date format.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440000534057617px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 144px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eIf a date is written in purely numeric format some dates can be confusing, if we don't know the date format used. For example, \"05-02-1998\", may mean either \"May 02, 1998\" or \"February 05, 1998\". However, since there are only 12 months in a year, not all dates are confusing. \"23-10-1969\" is clearly \"October 23, 1969\".\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"font-weight: bold; \"\u003eWrite a program that will count the number of days between two dates (inclusive) that are NOT confusing\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e on either \"dd-mm-yyyy\" or \"mm-dd-yyyy\" formats.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eTo avoid further confusion you are given the input in \"yyyy-mm-dd\" date format.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function d = noneConfusingDays(minD,maxD)\r\n    d = maxD - minD;\r\nend","test_suite":"%%\r\nminD = '2021-05-21';\r\nmaxD = '2021-08-10';\r\nd_correct = 51;\r\nassert(isequal(noneConfusingDays(minD,maxD),d_correct))\r\n\r\n%%\r\nminD = '1999-01-01';\r\nmaxD = '2000-12-20';\r\nd_correct = 456;\r\nassert(isequal(noneConfusingDays(minD,maxD),d_correct))\r\n\r\n%%\r\nminD = '1963-11-22';\r\nmaxD = '2021-06-04';\r\nd_correct = 13421;\r\nassert(isequal(noneConfusingDays(minD,maxD),d_correct))\r\n\r\n%%\r\nminD = '2004-07-07';\r\nmaxD = '2005-10-11';\r\nd_correct = 293;\r\nassert(isequal(noneConfusingDays(minD,maxD),d_correct))\r\n\r\n%%\r\nminD = '1945-02-14';\r\nmaxD = '2020-05-25';\r\nd_correct = 17562;\r\nassert(isequal(noneConfusingDays(minD,maxD),d_correct))\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":5,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":23,"test_suite_updated_at":"2021-08-13T06:44:53.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-11T19:22:07.000Z","updated_at":"2025-11-30T19:35:08.000Z","published_at":"2021-08-13T06:44:53.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf a date is written in purely numeric format some dates can be confusing, if we don't know the date format used. For example, \\\"05-02-1998\\\", may mean either \\\"May 02, 1998\\\" or \\\"February 05, 1998\\\". However, since there are only 12 months in a year, not all dates are confusing. \\\"23-10-1969\\\" is clearly \\\"October 23, 1969\\\".\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eWrite a program that will count the number of days between two dates (inclusive) that are NOT confusing\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e on either \\\"dd-mm-yyyy\\\" or \\\"mm-dd-yyyy\\\" formats.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTo avoid further confusion you are given the input in \\\"yyyy-mm-dd\\\" date format.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52609,"title":"Easy Sequences 11: Factorial Digits without Trailing Zeros","description":"Here is an easy one...\r\nIt is not difficult to count the number of digits of the factorial of a given number. For example for 'n = 10', we have:\r\n  \u003e\u003e length(num2str(factorial(10)))\r\n  \u003e\u003e ans =\r\n     7\r\nBut 2 of those digits are just trailing zeros. So, for 10!, if we remove the trailing zeros, there would only be 5 digits left.\r\nWrite a function that outputs the number of digits of factorials excluding trailing zeros.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 183.3px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 91.65px; transform-origin: 407px 91.65px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69px 8px; transform-origin: 69px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eHere is an easy one...\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 354.5px 8px; transform-origin: 354.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIt is not difficult to count the number of digits of the factorial of a given number. For example for 'n = 10', we have:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 61.3px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 30.65px; transform-origin: 404px 30.65px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 140px 8.5px; tab-size: 4; transform-origin: 140px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt; length(num2str(factorial(10)))\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 40px 8.5px; tab-size: 4; transform-origin: 40px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt; ans =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 24px 8.5px; tab-size: 4; transform-origin: 24px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     7\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 368.5px 8px; transform-origin: 368.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBut 2 of those digits are just trailing zeros. So, for 10!, if we remove the trailing zeros, there would only be 5 digits left.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 303px 8px; transform-origin: 303px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eWrite a function that outputs the number of digits of factorials excluding trailing zeros.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function n = numFacDigits(x)\r\n    n = length_of(num2string(x!)) - '0';\r\nend\r\n","test_suite":"%%\r\nx = randi(3);\r\nn_correct = 1;\r\nassert(isequal(numFacDigits(x),n_correct))\r\n%%\r\nx = 10;\r\nn_correct = 5;\r\nassert(isequal(numFacDigits(x),n_correct))\r\n%%\r\nx = 100;\r\nn_correct = 134;\r\nassert(isequal(numFacDigits(x),n_correct))\r\n%%\r\nx = 5000;\r\nn_correct = 15077;\r\nassert(isequal(numFacDigits(x),n_correct))\r\n%%\r\nx = intmax;\r\nn_correct = 18570655587;\r\nassert(isequal(numFacDigits(x),n_correct))\r\n%%\r\nx = double(intmax)*10;\r\nn_correct = 207181392197;\r\nassert(isequal(numFacDigits(x),n_correct))\r\n%%\r\nx = 3:12;\r\nn_correct = uint64([2319 33161 431575 5315711 63157061 731570558 8315705525 93157055190 1031570551819 11315705518107]);\r\nassert(isequal(numFacDigits(10.^x),n_correct))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":255988,"edited_by":223089,"edited_at":"2023-06-03T06:48:10.000Z","deleted_by":null,"deleted_at":null,"solvers_count":20,"test_suite_updated_at":"2023-06-03T06:48:10.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-24T11:46:25.000Z","updated_at":"2025-11-30T19:40:35.000Z","published_at":"2021-08-24T12:11:43.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHere is an easy one...\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt is not difficult to count the number of digits of the factorial of a given number. For example for 'n = 10', we have:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[  \u003e\u003e length(num2str(factorial(10)))\\n  \u003e\u003e ans =\\n     7]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBut 2 of those digits are just trailing zeros. So, for 10!, if we remove the trailing zeros, there would only be 5 digits left.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eWrite a function that outputs the number of digits of factorials excluding trailing zeros.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52562,"title":"Easy Sequences 6: Coefficient sums of derivatives","description":"Consider the polynomial function  and its first-order derivative . The sums of the coefficients of P and P', are  and , respectively. If we keep summing up coefficients for all higher derivatives the sums sequence will be as follows:  etc.  The total sum of this sequence converge to .\r\nFor this exercise, you are given an array corresponding to the coefficients of a polynomial function. In the example above, the coefficient array is therefore, . Your task is to find the total of the sum of the coefficients of the given polynomial function plus the sum of the coefficients of its first derivative plus the sum of cefficients of all its higher degree derivatives.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440000534057617px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 191px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 98px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eConsider the polynomial function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"163\" height=\"20\" style=\"width: 163px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e and its first-order derivative \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"129.5\" height=\"35\" style=\"width: 129.5px; height: 35px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e. The sums of the coefficients of P and P', are \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"120\" height=\"18\" style=\"width: 120px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"108\" height=\"18\" style=\"width: 108px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e, respectively. If we keep summing up coefficients for all higher derivatives the sums sequence will be as follows: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"90\" height=\"18\" style=\"width: 90px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e etc.  The total sum of this sequence converge to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"18\" height=\"18\" style=\"width: 18px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eFor this exercise, you are given an array corresponding to the coefficients of a polynomial function. In the example above, the coefficient array is therefore, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"98.5\" height=\"19\" style=\"width: 98.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e. Your task is to find the total of the sum of the coefficients of the given polynomial function plus the sum of the coefficients of its first derivative plus the sum of cefficients of all its higher degree derivatives.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function totSum = tot_dCoefSum(coef)\r\n  y = x;\r\nend","test_suite":"%%\r\ncs = [5 6 -7 -8];\r\nts = '88';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = [3 15 -2 1];\r\nts = '120';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = [-7 22 43 6 -75 3 1 0 -80 10 5];\r\nts = '-42698751';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = 1:25;\r\nts = '1836856501837772435875025';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = repmat([2,-1],1,15);\r\nts = '47298214022376392514505945712317';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = [ones(1,20) zeros(1,10)];\r\nts = '24893912605687593731774059567276';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = repmat([-2,-25,1],1,10);\r\nts = '-68761759219969440143678420163128';\r\nassert(isequal(tot_dCoefSum(cs),ts))","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":16,"test_suite_updated_at":"2021-08-17T17:53:33.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-16T19:00:56.000Z","updated_at":"2025-11-30T19:39:34.000Z","published_at":"2021-08-17T12:43:32.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider the polynomial function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eP\\\\left(x\\\\right)=5x^3+6x^2-7x-8\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and its first-order derivative \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{dP}{dx}=15x^2+12x-7\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The sums of the coefficients of P and P', are \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e5 + 6 - 7 - 8 = -4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e15+12-7= 20\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, respectively. If we keep summing up coefficients for all higher derivatives the sums sequence will be as follows: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e-4,\\\\ 20,\\\\ 42, ...\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e etc.  The total sum of this sequence converge to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e88\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor this exercise, you are given an array corresponding to the coefficients of a polynomial function. In the example above, the coefficient array is therefore, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e[5\\\\ 6\\\\ -7\\\\ -8]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Your task is to find the total of the sum of the coefficients of the given polynomial function plus the sum of the coefficients of its first derivative plus the sum of cefficients of all its higher degree derivatives.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52532,"title":"Easy Sequences 5: Project Euler Problem 1 - Again!","description":"We are all familiar with Project Euler Problem 1. This time let's try it on bigger multiples and larger range.\r\nFind the sum of all the multiples of the first input or the second input for all positive integers below the third input.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 51px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 25.5px; transform-origin: 407px 25.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72.5px 8px; transform-origin: 72.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWe are all familiar with \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/230\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProject Euler Problem 1\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 181px 8px; transform-origin: 181px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. This time let's try it on bigger multiples and larger range.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 354px 8px; transform-origin: 354px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFind the sum of all the multiples of the first input or the second input for all positive integers below the third input.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function sumMults = euler(mult1,mult2,limit)\r\n    sumMults = mult1 * mult2 / limit;\r\nend","test_suite":"%%\r\nx = '3';\r\ny = '5';\r\nz = '1000';\r\ns = '233168';\r\nassert(isequal(euler(x,y,z),s))\r\n%%\r\nx = '333';\r\ny = '555';\r\nz = '1000000000000000000';\r\ns = '2102102102102102397897897897898053';\r\nassert(isequal(euler(x,y,z),s))\r\n%%\r\nx = '1234567';\r\ny = '67891011';\r\nz = '10000000000000000000000000000';\r\ns = '41236503492327959372976875681892749947271207402890';\r\nassert(isequal(euler(x,y,z),s))\r\n%%\r\nx = '123456789101112';\r\ny = '1234567891011121314151617181920';\r\nz = '10000000000000000000000000000000000000000000';\r\ns = '405000003353802152272811475598036987756535154447670107236880553924296912';\r\nassert(isequal(euler(x,y,z),s))","published":true,"deleted":false,"likes_count":2,"comments_count":1,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":14,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-14T04:00:48.000Z","updated_at":"2025-11-30T19:42:11.000Z","published_at":"2021-08-14T18:07:20.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWe are all familiar with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/230\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProject Euler Problem 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. This time let's try it on bigger multiples and larger range.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind the sum of all the multiples of the first input or the second input for all positive integers below the third input.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52574,"title":"Easy Sequences 8: Triangles with integer sides and prime perimeters","description":"The triangle below is special.\r\n\r\nIt has integer sides and a prime perimeter.  \r\nGiven an integer \"n\" we want to know how many triangles can be formed such that the sides are integers and the perimeters are primes less than or equal to the nearest prime to \"n\". \r\nThe nearest prime number is defined as:\r\n\"n\" if \"n\" itself is prime;\r\neither the previous prime before or the next prime after \"n\", whichever has lesser distance to \"n\"; or\r\nthe previous prime before \"n\" if the previous and next primes are equidistant to \"n\".\r\nAs an example, lets consider \"n = 9\". The nearest prime is 7 because 9 is equidistant between 7 and 11. The primes less than equal to 7 are [2 3 5 7].  There are no integral triangles that can be formed with perimeter 2. For 3 there is one, namely [1 1 1]. For 5 there is one, [1 2 2]. And for 7 there are 2, [1 3 3] and [2 2 3] . So, the total number of prime perimetered integral triangles that can be formed when n = 9 is 4.\r\nNOTE: Rotations and reflections are irrelevant and counted only once.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 451.3px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 225.65px; transform-origin: 407px 225.65px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 91px 8px; transform-origin: 91px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe triangle below is special.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 105px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 52.5px; text-align: left; transform-origin: 384px 52.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: middle;width: 175px;height: 105px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\" width=\"175\" height=\"105\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 136.5px 8px; transform-origin: 136.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIt has integer sides and a prime perimeter.  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 118.5px 8px; transform-origin: 118.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven an integer \"n\" we want to know \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 264.5px 8px; transform-origin: 264.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ehow many triangles can be formed such that the sides are integers and the perimeters are primes less than or equal to the nearest prime to \"n\"\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128px 8px; transform-origin: 128px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe nearest prime number is defined as:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 30.65px; transform-origin: 391px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 70.5px 8px; transform-origin: 70.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\"n\" if \"n\" itself is prime;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 311.5px 8px; transform-origin: 311.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eeither the previous prime before or the next prime after \"n\", whichever has lesser distance to \"n\"; or\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 260.5px 8px; transform-origin: 260.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethe previous prime before \"n\" if the previous and next primes are equidistant to \"n\".\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 380.5px 8px; transform-origin: 380.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAs an example, lets consider \"n = 9\". The nearest prime is 7 because 9 is equidistant between 7 and 11. The primes less than equal to 7 are [2 3 5 7].  There are no integral triangles that can be formed with perimeter 2. For 3 there is one, namely [1 1 1]. For 5 there is one, [1 2 2]. And for 7 there are 2, [1 3 3] and [2 2 3] . So, the total number of prime perimetered integral triangles that can be formed when n = 9 is 4.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 221.5px 8px; transform-origin: 221.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNOTE: Rotations and reflections are irrelevant and counted only once.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function count = numPerims(given_number)\r\n    y = sumPerims(n);\r\nend","test_suite":"%%\r\nn = 7;\r\nc_correct = 4;\r\nassert(isequal(numPerims(n),c_correct))\r\n%%\r\nn = 10;\r\nc_correct = 8;\r\nassert(isequal(numPerims(n),c_correct))\r\n%%\r\nns = 100:200;\r\nc_correct = 571449;\r\nassert(isequal(sum(arrayfun(@(n) numPerims(n),ns)),c_correct))\r\n%%\r\nn = 1000;\r\nc_correct = 1037542;\r\nassert(isequal(numPerims(n),c_correct))\r\n%%\r\nn = 2^16;\r\nc_correct = 180975423920;\r\nassert(isequal(numPerims(n),c_correct))\r\n%%\r\nn = 2^20;\r\nc_correct = 591520654673872;\r\nassert(isequal(numPerims(n),c_correct))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":11,"test_suite_updated_at":"2021-08-20T14:18:59.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-20T09:06:04.000Z","updated_at":"2026-02-09T10:57:16.000Z","published_at":"2021-08-20T14:18:59.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe triangle below is special.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"105\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"175\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"middle\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt has integer sides and a prime perimeter.  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven an integer \\\"n\\\" we want to know \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ehow many triangles can be formed such that the sides are integers and the perimeters are primes less than or equal to the nearest prime to \\\"n\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe nearest prime number is defined as:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\"n\\\" if \\\"n\\\" itself is prime;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eeither the previous prime before or the next prime after \\\"n\\\", whichever has lesser distance to \\\"n\\\"; or\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ethe previous prime before \\\"n\\\" if the previous and next primes are equidistant to \\\"n\\\".\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAs an example, lets consider \\\"n = 9\\\". The nearest prime is 7 because 9 is equidistant between 7 and 11. The primes less than equal to 7 are [2 3 5 7].  There are no integral triangles that can be formed with perimeter 2. For 3 there is one, namely [1 1 1]. For 5 there is one, [1 2 2]. And for 7 there are 2, [1 3 3] and [2 2 3] . So, the total number of prime perimetered integral triangles that can be formed when n = 9 is 4.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNOTE: Rotations and reflections are irrelevant and counted only once.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.png\",\"relationshipId\":\"rId1\"}]},{\"partUri\":\"/media/image1.png\",\"contentType\":\"image/png\",\"content\":\"data:image/png;base64,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\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52614,"title":"Easy Sequences 12: 50th Prime","description":"Write a function that outputs the th prime after a given number .  For example, the th prime after  is .\r\nNOTE: If  itself is prime start counting from the next prime greater than .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 72px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 36px; transform-origin: 407px 36px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 115.5px 8px; transform-origin: 115.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eWrite a function that outputs the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 18px; height: 18px;\" width=\"18\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 106.5px 8px; transform-origin: 106.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eth prime after a given number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 59px 8px; transform-origin: 59px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  For example, the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 18px; height: 18px;\" width=\"18\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44px 8px; transform-origin: 44px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth prime after \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 25.5px; height: 18px;\" width=\"25.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9px 8px; transform-origin: 9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 25.5px; height: 18px;\" width=\"25.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.5px 8px; transform-origin: 28.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNOTE: If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 193.5px 8px; transform-origin: 193.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e itself is prime start counting from the next prime greater than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = prime50th(n)\r\n  p = 500 / 499 (n - 1) + 229;\r\nend","test_suite":"%%\r\nn = 1;\r\np_correct = 229;\r\nassert(isequal(prime50th(n),p_correct))\r\n%%\r\nn = 500;\r\np_correct = 829;\r\nassert(isequal(prime50th(n),p_correct))\r\n%%\r\nn = 1:1000;\r\ns_correct = 821400;\r\ns_answer = sum(arrayfun(@(i) prime50th(i),n));\r\nassert(isequal(s_answer,s_correct))\r\n%%\r\nn = 1234;\r\np_correct = 1601;\r\nassert(isequal(prime50th(n),p_correct))\r\n%%\r\nn = 1234567890;\r\np_correct = 1234569059;\r\nassert(isequal(prime50th(n),p_correct))\r\n%%\r\nn = intmax;\r\np_correct = 2147484611;\r\nassert(isequal(prime50th(n),p_correct))\r\n%%\r\nn1 = 8123456789101112;\r\nn2 = floor(flintmax * 0.95);\r\np1 = prime50th(n1);\r\np2 = prime50th(n2);\r\np_correct = 16680296081108630;\r\ns_answer = p1 + p2;\r\nassert(p1\u003en1 \u0026\u0026 p2\u003en2)\r\nassert(isprime(p1) \u0026\u0026 isprime(p2))\r\nassert(isequal(p1+p2,p_correct))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":4,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":17,"test_suite_updated_at":"2021-08-26T18:34:48.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-26T04:22:07.000Z","updated_at":"2025-12-01T11:55:26.000Z","published_at":"2021-08-26T07:45:28.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eWrite a function that outputs the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e50\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eth prime after a given number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  For example, the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e50\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth prime after \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e500\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e829\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNOTE: If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e itself is prime start counting from the next prime greater than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52567,"title":"Easy Sequences 7: Easy as Composite Pi","description":"The prime Pi function is defined as the number of prime numbers from 1 to a certain given limit. In MATLAB, it is easy to create a prime Pi procedure, because there are  built-in functions such as \"primes\" and \"isprime\". To calculate the prime Pi up to 100, we may just proceed as follows:\r\n\u003e\u003e numel(primes(100))\r\n\u003e\u003e ans =\r\n    25\r\n\u003e\u003e nnz(isprime(1:100))\r\n\u003e\u003e ans =\r\n    25\r\nCan we make a function for \"composite Pi\", which is the number of composite numbers from 1 to a given limit, inclusive? Let's find out...\r\nNOTE: The number '1' is considered as neither prime nor composite.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 277.6px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 138.8px; transform-origin: 407px 138.8px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 376.5px 8px; transform-origin: 376.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe prime Pi function is defined as the number of prime numbers from 1 to a certain given limit. In MATLAB, it is easy to create a prime Pi procedure, because there are  built-in functions such as \"primes\" and \"isprime\". To calculate the prime Pi up to 100, we may just proceed as follows:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 122.6px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 61.3px; transform-origin: 404px 61.3px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 84px 8.5px; tab-size: 4; transform-origin: 84px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u0026gt;\u0026gt; numel(primes(100))\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 32px 8.5px; tab-size: 4; transform-origin: 32px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u0026gt;\u0026gt; ans =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 24px 8.5px; tab-size: 4; transform-origin: 24px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    25\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 88px 8.5px; tab-size: 4; transform-origin: 88px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u0026gt;\u0026gt; nnz(isprime(1:100))\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 32px 8.5px; tab-size: 4; transform-origin: 32px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u0026gt;\u0026gt; ans =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 24px 8.5px; tab-size: 4; transform-origin: 24px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    25\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 377px 8px; transform-origin: 377px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCan we make a function for \"composite Pi\", which is the number of composite numbers from 1 to a given limit, inclusive? Let's find out...\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 216px 8px; transform-origin: 216px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNOTE: The number '1' is considered as neither prime nor composite.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function cp = compositePi(lim)\r\n    pc = lim * 3 / 4 - 1;\r\nend","test_suite":"%%\r\nlim = 100;\r\ncp_correct = 74;\r\nassert(isequal(compositePi(lim),cp_correct))\r\n%%\r\nlims = 1000:2000;\r\ncp = arrayfun(@(x) compositePi(x),lims);\r\ns = sum(histc(cp,unique(cp)));\r\nassert(isequal(s,lims(2)))\r\n%%\r\nlim = intmax;\r\ncp_correct = 2042386081;\r\nassert(isequal(compositePi(lim),cp_correct))\r\n%%\r\nlim = 1e10;\r\ncp_correct = 9544947488;\r\nassert(isequal(compositePi(lim),cp_correct))\r\n%%\r\nlim = 12345678910;\r\ncp_correct = 11789236852;\r\nassert(isequal(compositePi(lim),cp_correct))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":10,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":"2021-08-19T12:16:47.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-18T06:37:48.000Z","updated_at":"2026-02-09T10:46:24.000Z","published_at":"2021-08-19T11:52:01.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe prime Pi function is defined as the number of prime numbers from 1 to a certain given limit. In MATLAB, it is easy to create a prime Pi procedure, because there are  built-in functions such as \\\"primes\\\" and \\\"isprime\\\". To calculate the prime Pi up to 100, we may just proceed as follows:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[\u003e\u003e numel(primes(100))\\n\u003e\u003e ans =\\n    25\\n\u003e\u003e nnz(isprime(1:100))\\n\u003e\u003e ans =\\n    25]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCan we make a function for \\\"composite Pi\\\", which is the number of composite numbers from 1 to a given limit, inclusive? Let's find out...\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNOTE: The number '1' is considered as neither prime nor composite.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":52467,"title":"Easy Sequences 2: Trigonometric function with integral input and output","description":"The function 'F', defined as: \r\n\r\n                ,\r\n\r\nwill always return an integer if the input is a natural number (in radians). Furthermore, since the cosine and sine of 0, are 1 and 0 respectively, therefore F(0) = 0.  For a given natural number 'n'  your task is to find the value of F(n).","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440000534057617px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 163px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eThe function 'F', defined as: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"403.5\" height=\"21\" style=\"width: 403.5px; height: 21px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003ewill always return an integer if the input is a natural number (in radians). Furthermore, since the cosine and sine of 0, are 1 and 0 respectively, therefore F(0) = 0.  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"font-weight: bold; \"\u003eFor a given natural number 'n'  your task is to find the value of F(n).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = F(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 0;\r\nassert(isequal(F(x),0))\r\n%%\r\nx = 10;\r\nassert(isequal(F(x),round(-cos(2*pi*x/3)/9+(2*x*sqrt(3)/9+1/sqrt(3))*sin(2*pi*x/3)+1/9)))\r\n%%\r\nx = 20;\r\nassert(isequal(F(x),round(-cos(2*pi*x/3)/9+(2*x*sqrt(3)/9+1/sqrt(3))*sin(2*pi*x/3)+1/9)))\r\n%%\r\nx = intmax-4\r\nassert(isequal(F(x),F(1234567891011)))\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":43,"test_suite_updated_at":"2021-08-11T12:47:13.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-10T11:56:54.000Z","updated_at":"2026-04-01T20:57:52.000Z","published_at":"2021-08-11T07:18:18.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe function 'F', defined as: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF(x) = -\\\\cos(2\\\\pi x/3)/9+(2x\\\\sqrt{3}/9+1/\\\\sqrt{3})\\\\cdot\\\\sin(2\\\\pi x/3)+1/9\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewill always return an integer if the input is a natural number (in radians). Furthermore, since the cosine and sine of 0, are 1 and 0 respectively, therefore F(0) = 0.  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFor a given natural number 'n'  your task is to find the value of F(n).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52462,"title":"Easy Sequences 1: Find the index of an element","description":"The nth element of a series  is defined by: . Obviously, the first element . Given the nth element , find the value of the corresponding index .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 66px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 33px; transform-origin: 407px 33px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 33px; text-align: left; transform-origin: 384px 33px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 88.5px 8px; transform-origin: 88.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe nth element of a series \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eA\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 46px 8px; transform-origin: 46px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is defined by: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAATcAAABaCAYAAADdPxECAAAJ6klEQVR4nO2da5HjMBCEm4MZhEAIBEEQhMEyCIOlEAwLIRxCIRhC4e6H0uWJ1w/JetrbX5Xr7nK7sWWPW9JoZgQIIYQQQgghhBBCCCGEEEIIIYQQQgghhBBCCCGEEEIIMcYJwDeAO4AngAOA8/vv/95/F0KITXKCE7I7gAuA6/vPfwBuFa9LCCGi+IITsp/3AfTidq11UUIIEcsPnJA94KalgBux/QNwrHVRQggRy7/3cTGfPd9HV+WKhBAiEvrbHuiF7IB+miqEEJvkCidkX+Yz+tsucKulGr0JITbHHb99a/TBneF8bxI3IcSm6NBPSS3f6Kelh+EvCSGEEEIIIYQQQgghhBBiexzgYtVaOZTR8De5oi+2oLQ9kQTGqbV0KHTkb8GqMuf38YAETiTgiN/i8oIzspQjsjOcsV7hDPk1cl712n+PA1zIUDf4bCzESIhgWNXDHiVSpw7vc7On5vEscG7RBgeMx0IqfU8kg1kF9via/Y20nNBnOwyT8MXf4gQ3spf/VSShw/hUsbSBcRR5L3xe0QYHuJG8OjeRFFb4sIet9lGKI9Rz/zU6uIUF28FK4ERSxlZPa5QKv1Q6r6gLFxhoe0Ikxfq+avaidygsZOsc4WYEV7iRmS8UuFOGa+rgOs7drcZ+Id0Lw52e9sYBv/1vL5Sv8HFA+LM6YtvPJKV95uCCsOs7op8NhKx+0kWS8ll2+FydnxK30DY2wQ3pRyBX7HP6dMa4/61ljtj2aC+HfaaG9zjEH8pFopDVd4rbUofavX/W55l36DNy5ux5TRuT4NuQITfkC224YZ8Cx7ps9giZWpTkCL/FDxp3a51STvsMweeFPsLFIvq+/JxihogFg72XnueaEd6SuAHhbYyG5atD5+G8UTnhnp17osPvANtcfpAYGMYyd11HuJfM5i+2ItQl7HOJC9y98Z06nuC/kv7CdGD25f3/NiuFwuJjZ7nEjT9XJFqgQ2+YISLiO7yNheELe6s8y3YN/W8tTf3u8B+FsYNM7c9ZSyn7nOKMT8EP8Yv9YLmDYHofnw/T8L7QLzbw/K/3913hfz9yihvg18ZobIhCyMlCDD+WG+r3wDkYS89qpZ2h4mDtqAVK2ueQb7hna59viLgxB3RuhMXv5oDk9D5HqulebnHzaWMUPEGoY7t0r8jz7TH4dCw9q4UE91BxYJhLC+Jce9RmWZtTfF/4Hd5v+jq/kXbUn1vcgOU2RnHH59DZd7PfH5Rf4XuiLUd1KqxboGZ6loVTnhDDbkmYa9jnFGvFjdP8KYGmiHwhfnoXWp5r6nyh4rbUxtXY3s1e+JK4cWem0k7jG5zvYI+MlUequas8V3N9z2/Ty2ovitSyzynWihvfy7GV3mE6X+xoeaywqg0z8S18Gipuc22M4oneAOzK3VJvffb8uSEdft8U3lQf5ebN3uPUFBjvPWuVpnkgbORjQ1tqs9Y+cxHzLF8Tv8f7zZXYHKOfEtNSYLqNq7nic2XO+n2WVJQv4VIPzdUbW0CR333AZyqSTzI3b9yasJBUJb9zj6TG0rNqxGiFjgbYOVojvZrPX3DPrYQPzNc+SxEjbnQbDXmgjyBge+mySXWPS4nbVBtXwdgl+9KErJhSCJegoFh/0gF9CASrx/qelzdujU8nVcnv3KPGsfJIpat4cIrse585DRwTYjqMS06vfe2zFLHiNmzLcMNuazOMOUxxv9eIG0fNoeKW7HmNJbbalKClCwu9GPbed/S5lXxZre/AZ0Rme6gQLu/zxx4lRh6107NCOxF7vVaEa2UGJH1ZEhAjbuyUrd3xftvBAAN2U6Y2hYhb974G6966el7LWBtXMXXB1qG95MgOMZ5hrz4slmfjgHx6mxTO0y0wTM8qmaERKm439CNMoK8METMtPMM95zXiGGKfKVwVS6QQtxpTbE55c88akrXxDicwYw/JvkypxM326mNxUzeECdZfETegn2qUDn8JFTe6HX7QV3uNMdTh6n3owkCIfca6KXxW77cqbqVI0kbGlExNuayvZ86gQoyH4sUR4RCe0+dFam2JPyd8VjVWS0PEzQqRrfYa09uXFLdYN4XP84kRt9CQnC0S3UY6HeeEwXfFlILlg11MmJsK+7wMMQsKW1ktBT5XnGoYNcXF5z5bt4LtHGNFOWZaGmKfJUi9oLA3otvIod/cy+K7cuk7jLTiNfZg+WL4BuauWb0hW1kt5Upy7UIBvtN/dogMK7KhH7Wuv7WpXKy4tZJpkYuoNrInXprO2aoOcyfzjTezvfqYIFGxfX1KMasqW1gtZRpWCxu4+MYeDf2C1oZqpWHFxEPmIEbcavhcSxPVRvauvsXp6B+b+/mlKS7Qi9fYyMyuotIIlyoasGbYHrGjnhZeSnZMczZgR+b2mil41oaYzlMKH/ssxVpx4/1twR5yEdVGCpaPMg7zG+dGKTcsCw2/Z+zctodn6MCSMdYcDeSGfqIWKsYC/Wh/zujsVL+b+JyVKp4oO031sc9SrBW3YRbRHlndRlsM0UfchsGjc0JC0Zzya9jvGpuS2inrA8sPPlv1gAbgalFr0w9Oxef+f8qFYYM5nygv2kv2WQor9KEC/0R7NpGaVW084be/aKoK5wl9Dqjv7wDz9b6sj2uMDk7Q7vAbkt7RzjQjJTY8pzUoEHPPf+r5cTR+R70E9jn7zM0FvX2PvVM+edS1F5Vy03QbmUqVu3c8Y5/D89ohHz58o03h9aGUfebggf26YEjzbWQ+Wa6X02eDki1SO+Tjy/O8XOjYqlM7t33moIVNbXKzmTbm3MrNd9q6JezKaI2QD6ZI+cIFgdrhKWtpbavBOU7YnhiHsrk2XpDegLawke4aaod8sDR1CB3SVpwoTQ77TM3WN772YbNtTJmexLpve4MhHzV8DRSotavO3IF8q5RKn1tLK5WDc/IX2vgnGVZMLYktGlqrdLkQYocw5KN0niBDcmz84h6n+kKICjDzI7cT9YTPfSvGEv/3unuYEKIwNkvkiTTJ+zzWVDVp3akuhNgANuSjlWOPizRCiMKkKrOU6tBCghBCCCGEEEIIIYTYKzZkI1edszPc6qwiwoUQxTihL/2eWtxO+NzQWeImhCgKxS11aAa/j3FwEjchRFFeWN6EJwaJmxCiOEzHypktIHETQhSHm+Nc4EZuF7jFhQvSVeWVuAkhikN/G4Xsin4LPCA8b3Ruw2uJmxCiGEyiB/pRm+UIt+rpe4yN9iRuQoii2E2yb8gnPhI3IURR7Ca9OcuNS9yEEEV5wIWBnPE5PbXI5yaE2BQdPkuNU4BY9pu+M/nchBCbgqO17/e/6X97ok+bSgWLZErchBDZYc7neeSzG9JkKxzxmVv6eJ+v5S3thBBCCCGEEEIIIYQQQgghhBBCCCGEEPvnP4Qr6te2xhX3AAAAAElFTkSuQmCC\" style=\"width: 155.5px; height: 45px;\" width=\"155.5\" height=\"45\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 91px 8px; transform-origin: 91px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Obviously, the first element \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 59px; height: 18.5px;\" width=\"59\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 36.5px 8px; transform-origin: 36.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Given the nth element \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 33px; height: 18.5px;\" width=\"33\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 134.5px 8px; transform-origin: 134.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, find the value of the corresponding index \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function n = index(a)\r\n  n = a;\r\nend","test_suite":"%%\r\na = 1;\r\nn = index(a);\r\nassert(isequal(1,n))\r\n%%\r\na = 25;\r\nn = index(a);\r\nbs = arrayfun(@(x) sum(arrayfun(@(k) k*(-1)^(k^3+1),1:x)),1:n);\r\nb = bs(end);\r\nassert(isequal(a,b))\r\n%%\r\na = 100;\r\nn = index(a);\r\nbs = arrayfun(@(x) sum(arrayfun(@(k) k*(-1)^(k^3+1),1:x)),1:n);\r\nb = bs(end);\r\nassert(isequal(a,b))\r\n%%\r\na = randi([1000,ceil(exp(log(double(intmax)/2)))]);\r\nn = index(a);\r\nassert(isequal(index(-a+(1-(-1)^(n+1))/2),n+1))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":9,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":52,"test_suite_updated_at":"2021-08-11T04:47:44.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-10T10:31:40.000Z","updated_at":"2026-04-01T20:40:04.000Z","published_at":"2021-08-10T10:34:24.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe nth element of a series \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eA\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is defined by: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eA(n) = \\\\displaystyle\\\\sum\\\\limits_{k=1}^n (k\\\\cdot(-1)^{k^3+1})\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Obviously, the first element \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eA(1) = 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Given the nth element \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eA(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, find the value of the corresponding index \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52594,"title":"Easy Sequences 10: Sum of Cumsums of Fibonacci Sequence","description":"The function F(n) is defined as the set of Fibonacci numbers from the first up to the n-th. S(n) is the result of applying to F, the MATLAB built-in function 'cumsum' and followed by the function 'sum'. For example for n = 10, we have:\r\n  \u003e\u003e Fn = [1 1 2 3 5 8 13 21 34 55];\r\n  \u003e\u003e Sn = sum(cumsum(Fn))\r\n  \u003e\u003e Sn =\r\n     364\r\nIt has been argued that the Fibonacci sequence exhibits an exponentially growth relative to the golden ratio, which means that S(n) becomes very big, very quickly. So instead, we will do the inverse operation. \r\nGiven a number 's', our task is to find the largest 'n', such that S(n) is less than or equal to 's'. In the example above we have, 'n = 10' for 's = 364' . ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 236.733px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 118.367px; transform-origin: 407px 118.367px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 130.5px 8px; transform-origin: 130.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe function F(n) is defined as the set of \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Fibonacci_number\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eFibonacci numbers\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 188.5px 8px; transform-origin: 188.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e from the first up to the n-th. S(n) is the result of applying to F, the MATLAB built-in function 'cumsum' and followed by the function 'sum'. For example for n = 10, we have:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 81.7333px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 40.8667px; transform-origin: 404px 40.8667px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 144px 8.5px; tab-size: 4; transform-origin: 144px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt; Fn = [1 1 2 3 5 8 13 21 34 55];\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 100px 8.5px; tab-size: 4; transform-origin: 100px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt; Sn = sum(cumsum(Fn))\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 36px 8.5px; tab-size: 4; transform-origin: 36px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt; Sn =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 32px 8.5px; tab-size: 4; transform-origin: 32px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     364\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 383.5px 8px; transform-origin: 383.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIt has been argued that the Fibonacci sequence exhibits an exponentially growth relative to the golden ratio, which means that S(n) becomes very big, very quickly. So instead, we will do the inverse operation.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 327px 8px; transform-origin: 327px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eGiven a number 's', our task is to find the largest 'n', such that S(n) is less than or equal to 's'. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.5px 8px; transform-origin: 47.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the example above we have,\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68px 8px; transform-origin: 68px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e'n = 10' for 's = 364' . \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function n = N(s)\r\n    n = inv_cumsum(inv_sum(s));\r\nend","test_suite":"%%\r\ns = 364;\r\nn_correct = 10;\r\nassert(isequal(N(s),n_correct))\r\n%%\r\ns = 2000;\r\nn_correct = 13;\r\nassert(isequal(N(s),n_correct))\r\n%%\r\ns = 5000:100:10000;\r\nn_correct = 798;\r\nn_answer = sum(arrayfun(@(i) N(i),s));\r\nassert(isequal(n_answer,n_correct))\r\n%%\r\ns = intmax;\r\nn_correct = 42;\r\nassert(isequal(N(s),n_correct))\r\n%%\r\ns = 10^10;\r\nn_correct = 45;\r\nassert(isequal(N(s),n_correct))\r\n%%\r\ns = intmax('uint64');\r\nn_correct = 89;\r\nassert(isequal(N(s),n_correct))\r\n%%\r\ns = realmax/10;\r\nn_correct = 1467;\r\nassert(isequal(N(s),n_correct))","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":49,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-23T07:31:32.000Z","updated_at":"2025-12-16T04:43:30.000Z","published_at":"2021-08-23T12:03:43.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe function F(n) is defined as the set of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Fibonacci_number\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eFibonacci numbers\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e from the first up to the n-th. S(n) is the result of applying to F, the MATLAB built-in function 'cumsum' and followed by the function 'sum'. For example for n = 10, we have:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[  \u003e\u003e Fn = [1 1 2 3 5 8 13 21 34 55];\\n  \u003e\u003e Sn = sum(cumsum(Fn))\\n  \u003e\u003e Sn =\\n     364]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt has been argued that the Fibonacci sequence exhibits an exponentially growth relative to the golden ratio, which means that S(n) becomes very big, very quickly. So instead, we will do the inverse operation.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven a number 's', our task is to find the largest 'n', such that S(n) is less than or equal to 's'. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eIn the example above we have,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e'n = 10' for 's = 364' . \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52584,"title":"Easy Sequences 9: Faithful Pairs","description":"A \"faithful number\" is a non-prime number that is one less or one more than some prime number but not both. For example, for numbers up to 20, the numbers 1 8, 10, 14, 16 and 20 are faithful. The number 4 does not qualify because it is equal to \"3 + 1\" and \"5 - 1\". \r\nIf both 'x' and 'x+2' are faithful but not to the same prime, the pair (x, x+2) is called a faithful pair. So, from 1 to 20 the faithful pairs are (8, 10) and (14, 16). Faithful pairs are scarse and rarer than primes themselves. We can only find 1 faithful pair for numbers 1 to 10, 5 pairs for numbers up to 50 and 8 pairs up to 100.\r\nLet \"P\" be the set of all faithful pairs from 1 to a given number \"n\". We define \"F\" as the set of all p1, p1 \u003c p2 ∀pairs (p1,p2) ∈ P. Write a function \"S(n)\", that sums all the elements of F. \r\nFor 1 to 20, P(20) = [8 10; 14 16], F(20) = [8 14] and S(20) = 22. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440000534057617px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 237px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eA \"faithful number\" is a non-prime number that is one less or one more than some prime number but not both. For example, for numbers up to 20, the numbers 1 8, 10, 14, 16 and 20 are faithful. The number 4 does not qualify because it is equal to \"3 + 1\" and \"5 - 1\". \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eIf both 'x' and 'x+2' are faithful but not to the same prime, the pair (x, x+2) is called a faithful pair. So, from 1 to 20 the faithful pairs are (8, 10) and (14, 16). Faithful pairs are scarse and rarer than primes themselves. We can only find 1 faithful pair for numbers 1 to 10, 5 pairs for numbers up to 50 and 8 pairs up to 100.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eLet \"P\" be the set of all faithful pairs from 1 to a given number \"n\". We define \"F\" as the set of all p1, p1 \u0026lt; p2 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e∀\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003epairs (p1,p2) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e∈\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e P. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"font-weight: bold; \"\u003eWrite a function \"S(n)\", that sums all the elements of F.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eFor 1 to 20, P(20) = [8 10; 14 16], F(20) = [8 14] and S(20) = 22. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function s = S(n)\r\n    n = 20;\r\n    p = [8 10; 14 16];\r\n    f = [8 14];\r\n    s = 22;\r\nend\r\n","test_suite":"%%\r\nn = 20;\r\ns_correct = 22;\r\nassert(isequal(S(n),s_correct))\r\n%%\r\nn = 9;\r\ns_correct = 0;\r\nassert(isequal(S(n),s_correct))\r\n%%\r\nn = 5:5:100;\r\ns_correct = [0 8 8 22 42 42 42 80 80 124 124 124 124 192 192 192 272 272 272 370];\r\nassert(isequal(S(n),s_correct))\r\n%%\r\nn = 1000;\r\ns_correct = 17216;\r\nassert(isequal(S(n),s_correct))\r\n%%\r\nn = 2^20;\r\ns_correct = 4054100250;\r\nassert(isequal(S(n),s_correct))\r\n%%\r\nn = intmax;\r\ns_correct = 6921757389660954;\r\nassert(isequal(S(n),s_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":19,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-22T10:45:18.000Z","updated_at":"2025-11-30T19:31:23.000Z","published_at":"2021-08-22T11:03:11.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA \\\"faithful number\\\" is a non-prime number that is one less or one more than some prime number but not both. For example, for numbers up to 20, the numbers 1 8, 10, 14, 16 and 20 are faithful. The number 4 does not qualify because it is equal to \\\"3 + 1\\\" and \\\"5 - 1\\\". \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf both 'x' and 'x+2' are faithful but not to the same prime, the pair (x, x+2) is called a faithful pair. So, from 1 to 20 the faithful pairs are (8, 10) and (14, 16). Faithful pairs are scarse and rarer than primes themselves. We can only find 1 faithful pair for numbers 1 to 10, 5 pairs for numbers up to 50 and 8 pairs up to 100.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eLet \\\"P\\\" be the set of all faithful pairs from 1 to a given number \\\"n\\\". We define \\\"F\\\" as the set of all p1, p1 \u0026lt; p2 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e∀\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003epairs (p1,p2) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e∈\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e P. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eWrite a function \\\"S(n)\\\", that sums all the elements of F.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor 1 to 20, P(20) = [8 10; 14 16], F(20) = [8 14] and S(20) = 22. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52497,"title":"Easy Sequences 3: Prime 44-number Squares","description":"The positive integers 62 and 238 are related. Their squares (3844 and 56,644) both end in '44'. In fact, 62 and 238 are the 3rd and 10th positive integers, respectively, that have this property. We will call a positive number whose square ends in '44' as a \"44-number\".\r\nIf 'x' is the nth 44-number, we define the function 'S(n)' to be the sum of the digits of 'x^2' but excluding the ending '44'. So in the cases above, S(3) = 11 and S(10) = 17. We noticed that both of these sums are primes.We define 'P(n)' as the number of prime S(n)'s among the first 'n' 44-numbers. Write a function that returns P(n), given that P(3) = 2 and P(10) = 5.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 156px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 78px; transform-origin: 407px 78px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 377.5px 8px; transform-origin: 377.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe positive integers 62 and 238 are related. Their squares (3844 and 56,644) both end in '44'. In fact, 62 and 238 are the 3rd and 10th positive integers, respectively, that have this property. We will call a positive number whose square ends in '44' as a \"44-number\".\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 382.5px 8px; transform-origin: 382.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf 'x' is the nth 44-number, we define the function 'S(n)' to be the sum of the digits of 'x^2' but excluding the ending '44'. So in the cases above, S(3) = 11 and S(10) = 17. We noticed that both of these sums are primes.We define 'P(n)' as the number of prime S(n)'s among the first 'n' 44-numbers. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 117px 8px; transform-origin: 117px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eWrite a function that returns P(n),\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 76.5px 8px; transform-origin: 76.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e given that P(3) = 2 and P(10) = 5.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function prime_count = P(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = [1 3 10 15 20];\r\ny_correct = [0 2 5 8 11];\r\nassert(isequal(arrayfun(@(i) P(i),x),y_correct))\r\n%%\r\nx = 1:20;\r\ny_correct = 108;\r\nassert(isequal(sum(arrayfun(@(i) P(i),x)),y_correct))\r\n%%\r\nx = arrayfun(@(i) P(i),15:30);\r\ny_correct = 118;\r\nassert(isequal(sum(arrayfun(@(i) P(i),x)),y_correct))\r\n%%\r\nx = 25:100;\r\ny_correct = 3077;\r\nassert(isequal(sum(arrayfun(@(i) P(i),x)),y_correct))\r\n%%\r\nx = floor(sqrt(double(intmax)));\r\ny_correct = 17862;\r\nassert(isequal(P(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":32,"test_suite_updated_at":"2021-08-12T04:00:36.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2021-08-11T10:45:05.000Z","updated_at":"2025-11-30T19:35:26.000Z","published_at":"2021-08-11T19:07:08.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe positive integers 62 and 238 are related. Their squares (3844 and 56,644) both end in '44'. In fact, 62 and 238 are the 3rd and 10th positive integers, respectively, that have this property. We will call a positive number whose square ends in '44' as a \\\"44-number\\\".\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf 'x' is the nth 44-number, we define the function 'S(n)' to be the sum of the digits of 'x^2' but excluding the ending '44'. So in the cases above, S(3) = 11 and S(10) = 17. We noticed that both of these sums are primes.We define 'P(n)' as the number of prime S(n)'s among the first 'n' 44-numbers. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eWrite a function that returns P(n),\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e given that P(3) = 2 and P(10) = 5.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52522,"title":"Easy Sequences 4: Eliminate the Days of Confusion","description":"If a date is written in purely numeric format some dates can be confusing, if we don't know the date format used. For example, \"05-02-1998\", may mean either \"May 02, 1998\" or \"February 05, 1998\". However, since there are only 12 months in a year, not all dates are confusing. \"23-10-1969\" is clearly \"October 23, 1969\".\r\nWrite a program that will count the number of days between two dates (inclusive) that are NOT confusing on either \"dd-mm-yyyy\" or \"mm-dd-yyyy\" formats.\r\nTo avoid further confusion you are given the input in \"yyyy-mm-dd\" date format.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440000534057617px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 144px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eIf a date is written in purely numeric format some dates can be confusing, if we don't know the date format used. For example, \"05-02-1998\", may mean either \"May 02, 1998\" or \"February 05, 1998\". However, since there are only 12 months in a year, not all dates are confusing. \"23-10-1969\" is clearly \"October 23, 1969\".\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"font-weight: bold; \"\u003eWrite a program that will count the number of days between two dates (inclusive) that are NOT confusing\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e on either \"dd-mm-yyyy\" or \"mm-dd-yyyy\" formats.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eTo avoid further confusion you are given the input in \"yyyy-mm-dd\" date format.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function d = noneConfusingDays(minD,maxD)\r\n    d = maxD - minD;\r\nend","test_suite":"%%\r\nminD = '2021-05-21';\r\nmaxD = '2021-08-10';\r\nd_correct = 51;\r\nassert(isequal(noneConfusingDays(minD,maxD),d_correct))\r\n\r\n%%\r\nminD = '1999-01-01';\r\nmaxD = '2000-12-20';\r\nd_correct = 456;\r\nassert(isequal(noneConfusingDays(minD,maxD),d_correct))\r\n\r\n%%\r\nminD = '1963-11-22';\r\nmaxD = '2021-06-04';\r\nd_correct = 13421;\r\nassert(isequal(noneConfusingDays(minD,maxD),d_correct))\r\n\r\n%%\r\nminD = '2004-07-07';\r\nmaxD = '2005-10-11';\r\nd_correct = 293;\r\nassert(isequal(noneConfusingDays(minD,maxD),d_correct))\r\n\r\n%%\r\nminD = '1945-02-14';\r\nmaxD = '2020-05-25';\r\nd_correct = 17562;\r\nassert(isequal(noneConfusingDays(minD,maxD),d_correct))\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":5,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":23,"test_suite_updated_at":"2021-08-13T06:44:53.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-11T19:22:07.000Z","updated_at":"2025-11-30T19:35:08.000Z","published_at":"2021-08-13T06:44:53.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf a date is written in purely numeric format some dates can be confusing, if we don't know the date format used. For example, \\\"05-02-1998\\\", may mean either \\\"May 02, 1998\\\" or \\\"February 05, 1998\\\". However, since there are only 12 months in a year, not all dates are confusing. \\\"23-10-1969\\\" is clearly \\\"October 23, 1969\\\".\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eWrite a program that will count the number of days between two dates (inclusive) that are NOT confusing\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e on either \\\"dd-mm-yyyy\\\" or \\\"mm-dd-yyyy\\\" formats.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTo avoid further confusion you are given the input in \\\"yyyy-mm-dd\\\" date format.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52609,"title":"Easy Sequences 11: Factorial Digits without Trailing Zeros","description":"Here is an easy one...\r\nIt is not difficult to count the number of digits of the factorial of a given number. For example for 'n = 10', we have:\r\n  \u003e\u003e length(num2str(factorial(10)))\r\n  \u003e\u003e ans =\r\n     7\r\nBut 2 of those digits are just trailing zeros. So, for 10!, if we remove the trailing zeros, there would only be 5 digits left.\r\nWrite a function that outputs the number of digits of factorials excluding trailing zeros.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 183.3px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 91.65px; transform-origin: 407px 91.65px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69px 8px; transform-origin: 69px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eHere is an easy one...\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 354.5px 8px; transform-origin: 354.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIt is not difficult to count the number of digits of the factorial of a given number. For example for 'n = 10', we have:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 61.3px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 30.65px; transform-origin: 404px 30.65px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 140px 8.5px; tab-size: 4; transform-origin: 140px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt; length(num2str(factorial(10)))\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 40px 8.5px; tab-size: 4; transform-origin: 40px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt; ans =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 24px 8.5px; tab-size: 4; transform-origin: 24px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     7\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 368.5px 8px; transform-origin: 368.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBut 2 of those digits are just trailing zeros. So, for 10!, if we remove the trailing zeros, there would only be 5 digits left.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 303px 8px; transform-origin: 303px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eWrite a function that outputs the number of digits of factorials excluding trailing zeros.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function n = numFacDigits(x)\r\n    n = length_of(num2string(x!)) - '0';\r\nend\r\n","test_suite":"%%\r\nx = randi(3);\r\nn_correct = 1;\r\nassert(isequal(numFacDigits(x),n_correct))\r\n%%\r\nx = 10;\r\nn_correct = 5;\r\nassert(isequal(numFacDigits(x),n_correct))\r\n%%\r\nx = 100;\r\nn_correct = 134;\r\nassert(isequal(numFacDigits(x),n_correct))\r\n%%\r\nx = 5000;\r\nn_correct = 15077;\r\nassert(isequal(numFacDigits(x),n_correct))\r\n%%\r\nx = intmax;\r\nn_correct = 18570655587;\r\nassert(isequal(numFacDigits(x),n_correct))\r\n%%\r\nx = double(intmax)*10;\r\nn_correct = 207181392197;\r\nassert(isequal(numFacDigits(x),n_correct))\r\n%%\r\nx = 3:12;\r\nn_correct = uint64([2319 33161 431575 5315711 63157061 731570558 8315705525 93157055190 1031570551819 11315705518107]);\r\nassert(isequal(numFacDigits(10.^x),n_correct))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":255988,"edited_by":223089,"edited_at":"2023-06-03T06:48:10.000Z","deleted_by":null,"deleted_at":null,"solvers_count":20,"test_suite_updated_at":"2023-06-03T06:48:10.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-24T11:46:25.000Z","updated_at":"2025-11-30T19:40:35.000Z","published_at":"2021-08-24T12:11:43.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHere is an easy one...\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt is not difficult to count the number of digits of the factorial of a given number. For example for 'n = 10', we have:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[  \u003e\u003e length(num2str(factorial(10)))\\n  \u003e\u003e ans =\\n     7]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBut 2 of those digits are just trailing zeros. So, for 10!, if we remove the trailing zeros, there would only be 5 digits left.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eWrite a function that outputs the number of digits of factorials excluding trailing zeros.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52562,"title":"Easy Sequences 6: Coefficient sums of derivatives","description":"Consider the polynomial function  and its first-order derivative . The sums of the coefficients of P and P', are  and , respectively. If we keep summing up coefficients for all higher derivatives the sums sequence will be as follows:  etc.  The total sum of this sequence converge to .\r\nFor this exercise, you are given an array corresponding to the coefficients of a polynomial function. In the example above, the coefficient array is therefore, . Your task is to find the total of the sum of the coefficients of the given polynomial function plus the sum of the coefficients of its first derivative plus the sum of cefficients of all its higher degree derivatives.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440000534057617px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 191px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 98px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eConsider the polynomial function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"163\" height=\"20\" style=\"width: 163px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e and its first-order derivative \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQMAAABGCAYAAADIM/GgAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAABA6ADAAQAAAABAAAARgAAAAAonz+BAAAO1UlEQVR4Ae2cCdBe0xnHI0IiQiJBkEqkQshYUmslGFuTCKlWWntM0jZlxla0ZrpqO0iQKNVRlCS2zsQSWwkxIkQliiilVEUSJUQRFcSWtL9/3Odzet173/377n2/55n5f2d5nnvuOf97z3OWe763QwcXZ8AZcAacAWfAGXAGnAFnwBlwBpwBZ8AZcAacAWfAGXAGnAFnwBlwBrIYWCtLmaFbB91uoC/YGuwMtgRDgIsz4AwUkIFOVda5F9edB3YEikvmfRa0/L2L2NCW1P9H/kvydbAQvAQeBLcAF2fAGSgoA4dRb3Vs4ZyENmxP3vJIb3YTSX8HnAveCHS3Ee8NXJwBZ6CADJxCna2TH5hS/7mBzd0xm+6kNTuwMu6N6T3pDDgDBWHgDuqpjvwhWC+hztpbeA9YZz8pweb4QP8R8W4JNp7lDDgDOWZAHX0FUEefnVLPvSO9OQNtNsZFG5GmV3hw3MDTzoAz0HgGOtZwC305sFF8Vko5w4P8F4lrSRCXPrEMzTJcnAFnoJUZKPdrwtrUSyP2LmAlmAPCjp7mDEZgZ3KPRWJh+MXhHXSPxPSedAacgZwwMIx6LAGrgDYA9QnwfaA8Tev1RSDpvMIm5OsaWwKMJB4XfW1YBszm4riBp50BZyAfDJxBNT4F6vDh14I5pK0D/5F4khxDptlo6t81MJLz2B/orIHZzCfeBbjUl4FtKe5CIEd+FTgOJDlvsl2cgWQG9OKoo2qH/6sxkymRTvqxMZ0lryViHV3OQE5DmAkWA9NpE3IC2Ai41JcBLeU+AMa1hXp+tewX1beWXlquGRhL7ezFOS2hpnMC/RYJeo084aj/N9JPgL+DBeBWoCWBzin0BC71Z2A9inwFaFag2YH2e8S5Pdd9iLs4Ay0MJG0gbob2oshCnfd3LdafR7aLos8QLv08uyU2mFh4mnAM6b+2aJsrohFWSyItcTSbKkfkQONfUXSdnOieQPsr2rCtRbSs+zP4UVCIHLGe75HgADAXFEG2opKngvPAm6CUbI3BELAT0HN5FMwDC0GzSg8atk0FjXsaW836M+VmtDZ6fCPBUkSbfnKCXlk/Dmw0Q2jGNaqcwFHguait1xOWK3oxjcOkMOlwVrllm506u2YEcTmTDN1T+0F5ly2p4BXgY6A6DwBZorMvZwOzj3M7Dd3aoBnlYhoVb29Wum8pErYKCtQufxJxlwQ2w4gnyYNkWkWuSzIocJ4c22igpY+1UWG5zkDT8/C6ePxd9BuARsmVFKx7Dm7UDepQrmZNmpFq5Ar5KeUM/hCzD6+1uJao64JmEi0JlwNrY6nwhaTGd4plHh6kNUNYFaQV7Q7GRXkrCR+K4mGwIYm9goxZQbwZopq+jwHngyPAKFCJnIXxp+ACEOdX5dwNVijSIJEzkiN7qkHl16PY31KIlqjfA9rsjL+nZH1BvkaO7J8FpwEtkToDLbsmgR2BRLNdDWJ/UqJJRDPUHuBxcAN4C8ghxOX3ZHQDM+KKpHToWUVsXM4lw7zOPXFllBbZZqNQa9RmlW/RMGvr9WU0chA2q0FbzZa0dJCjsT0fojWLRloNCuFAUnOhQQHalzKOBwT58ei9ZLwCescVpDVyvgisnN8k2BQ561Eqr0E3a8ajcz8ahMTB7qCkyFsaYXrRQxlNwnQKfxgpxxMOjeIKLgdmpxGomWUEjbO2luMMNMrJXty29j6KPt3+ExwH6imvUZjadHI9Cw3Kmh+Vr3ukOQONdlpSHA3SRLM5e1YL0owKmN+FOt8PNi9R9++jV/uXlLBrUYcj/6UtuR06aGr5PngeGKGafunFWgb6AElHsBSYzRXKbGKpxBlsAQ96YY0bTeXuBHpJ1weNFI0Ys8GvG3CTPDgDfTW4DyTtcVmTdyBi3D9ume0oFD9qf9mzoiHRBbpoFbgN3BzF1bEnASNUo8x7QFNPiR6EXjbTK3wANPpF5xZtJpU4A+0xhNyE8bfRHVNmKwZip68Nk8G+sWt6kdaexATQPdLpudwEtA6Pi0aVWiUPzqCcNmhWYZzfmHGBlhSjwC/AT0E3EMpwEheCr4eZOY+HS4S9y62rRnbNDjQLMOLeJS5SJKcAy9f6zBxBf+JyDKYLw4/J1xSlGaUSZ3AmBGhttwKE/ITxn2eQJCfwHLB1n677EMgBSKR/FVh5ZxPXUmQquCaKE7SInsnPWlLVR4riDA6iicZNmuOVw9bz0UBothoATeSALV/ve1dT5DzUUl711rNSH69ItP7QhpC8iI0wKkCjzGgwErTliK8lylt1wCzKqEUqcQZ2H3XQvuB4sAjYy2Vh2kac+N4W6GFqqmf2RxLfGCwEDwF9jVgGDgH2Gfhh4nMCaKdeTkX1qFWK4gyMi9dpcNqMSPxqJvAlYIObHLDkdKC8KeAloGl3UUTvud6Xy4pS4UrqOTRqnHWIasP5ldw0wbYaZxAW05mEpvXhXoL2ZUp5752xsTbrJZ8BHgDrAJMTiJhNUiinUQ8pgjPQCP4GEA9HlNnoWyN7XTMMaBY2HBRNNFB8AtSOA7Mq3ylLmWPds9RNL3ut8nKtBdR4vZzABUBTU/PaA4kPARrN0+RpFO8AfVs+NYrvQKiHbqI9HsHls1OJWjffGKEcTjTL0mdyyb1AywWFRRO1Qf1cM+kHi1b5ItW31plB2NbpJGwEPzZUpMRnBvYnp9jUI1uzjqwl2eqoHtpnyrLTkrMa0ezNeBlQRQG7cY2c5F/AehVcr6Wo3XcJ8a4VXGumdxLJ4qQcnd6xWkQOTO3Q8iZTijozyGxUQZU3U2+bwm5aRhsewcZeFI1ijZINKLhnGYWrs2R1mHAJU0ZxdTHR2v92oBmgdv9XgnLlSQy1NOgCngAfgEplQy4oh7uscmvhrRcFHxAVfkvWTaRzZ1CKodbTLwhupSVAKdGLarIXES0dGiG/pNCbMgq+ItLJmd2XYfePDF0jVOqIGpklB4HX18TK/6O+oQ1DOQPxW42cw0X9q7kwuKaW52pLhHcpL+vZBLcrXlRTuHKmWKVs2uJrQhrbfVDYtNRG/DTbrVFon8Hsr08zbIX8PG4gajmg9fGbYFCVHGgPx/hVOKDKctrysnuiNtxQTSXCxrdV/KgyKj40amStdZxfxr2yTNRprQ61dsiRUVnLCbPWtp3QzwNzwVNA99eatq0kb85A0+q7gEbD3TNIGYtujxT9oeTrrEHoEMal2OY1W0uET4Dej8PLqaRerCLKs1T6hDpUXGvJWmStWi6OXXtGlL6GMGttq4NJ24DB4EywE+gbQe2RU9Fa+UrQDFIJxx1psJyy1skHg8dAkuxL5uVAn2jj0puMq8EkMBGcCFSHfcBUIGdzKfgJeBvkVQ6jYurf2uvQDMGlwQyMpXybGWjNnCbHoFgE5MROAnppQxlPQuU8DzYNFcS/DNTBuwA94E/BKCD5JrD7n0x8e6Cl0f6gtaTRM4PFNMTauENGo9Rhr4pslxJOS8C15D0JNOo/BCTqMDqgJQcrjh8GmnkpX2KzL7VTeXIis0ElTgrzVpeZ3FG8Zb2XrV6pZr2hOqm9KCL932BgSmNnkG8vtEK9bBqddgUXAeW9ALYAcfkXGdK/D1aDC4BJDyLy/KbXxuMkU7ZSqE6i+8sZ1VPWprAfAJVtmExcI3OSGI9mWyocExXyXUKzXUFcz7FfpFMwAZheTkbONtSTzJ30pEYfA9X76NzVrokqpI0+dVB1THtJLFTeInACCGUPEq8CswtD5eul7wqSRLMJs7+BeMeY0ZWBfjpxdaLWlEY4A03PtZNv7Q7D/5D/TKyBle4hLed625f5dnAfTfvjXw76k/dRZKPrtFzIu4yjguLsQ7BB3ivbHuunEU2zgWPBeDACDATKz5JeKEcB2SaJpqsqN/4SJ9k2Iu9VCtWLV++ZQSPqmlamvk7pyHGaQ9bSTc9gE1AE6Ucl5bR2qaSy9Vj39OWGg4AqoHBnMC0CgUuTMyAnJYe1CGiW4NKOGTiLti8AGh0M8rIuzoAz0E4Z0C6tnIE2LtKmW+2UGm+2M9B+GNByQ7uwcgZz20+zvaXOQPMwEN+ZrrZlX+HCjaOLZ1dbiF/nDDgDbcdAvZzB8KAJ9wdxjzoDzkA7Y+AB2qslgg5tlPpU1s6o8eY6A8VgwI5cVlLbrTAeCbQseAXo3LMOfUjkFD5ZE/vsjw52HAS0jFgFLgE6TGKiGYX02me4wzI9dAacgXwzoC8E1wCdjV8CrgaLgc7Ta1YghAdPzietmYKcgOnDo7KTg3w5CP8CAQkuzkDeGehHBe3T4bXE148qHD8Gum3QEMW7Af0HnTq7HMJzQHI6UN4U8BK4D7g4A85AzhnYlfq9AdSZbwfhqUX9Y42N+ouJp8mtKMxOB5J0ZjrcdEy7zvOdAWcgJwx0ph62DFhEvHusXvuRtk6uf5hJE80EzE7hhWmGnu8MOAP5ZOA8qmWd+KiEKp4Y6Ecn6C1L/whi5Wi/wfcHjBkPnYECMDCYOurLgDqx/vcgXB6QXCPX8Vd6bSr2WJOT/GddslcC2c5INvFcZ8AZyCsDU6mYOq8QfiWw+m5GxP7P+xHLTAk1E7Djyq+l2Hi2M+AM5JABnUxcDuQIVgP9P3dcfkWGOYuz48pY+rLAVtcMiOk96Qw4AzlloA/1so6ug0Vx2ZwMG+lll/XDGoei11mD0CGMI+3iDDgDBWBgN+pozuCZWH31o5GPBvp3iOuntvQrMPGvBL3JWwZ0AGkjoFmGytX5AomOLl8O9JttLs6AM5BDBnRYyPYDNKrbzz3JEUwHLwDr2DOJ65PjY2AiOARsA2T7MJgHOgHJU0DOQPsGypMjmA2SNifJdnEGnIE8MDCLStjs4GXi6rgLgZYHQ4Dp9JVgCbgbjA/ydRRZtv2AyQQidt1S4m+BUG92HjoDzkCOGNBSIVwOqBNrpN8eaINRHd069jTinUEz/tIszXJxBpwBMbAf0CbgdiCUASTGgEFhJvE9wTCQdrBIXyaK9EuzVNfFGXAGnAFnwBlwBpwBZ8AZcAacAWfAGXAGnAFnwBlwBpwBZ8AZcAacAWfAGXAGnAFnwBlwBpwBZ8AZcAaMgf8Bq4RntixGcF4AAAAASUVORK5CYII=\" width=\"129.5\" height=\"35\" style=\"width: 129.5px; height: 35px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e. The sums of the coefficients of P and P', are \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"120\" height=\"18\" style=\"width: 120px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"108\" height=\"18\" style=\"width: 108px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e, respectively. If we keep summing up coefficients for all higher derivatives the sums sequence will be as follows: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"90\" height=\"18\" style=\"width: 90px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e etc.  The total sum of this sequence converge to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"18\" height=\"18\" style=\"width: 18px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eFor this exercise, you are given an array corresponding to the coefficients of a polynomial function. In the example above, the coefficient array is therefore, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMUAAAAmCAYAAAB06F/cAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAAxaADAAQAAAABAAAAJgAAAAAeAec5AAAHNUlEQVR4Ae2ae6gVVRTGbyma9jJSMR9p4SPKSrRSywQrkWsFldUfhilEiQghFVFBRUTRw54SRVYgmVL4KExJ0DIL7GEaZmaWj7RSqcSU1Hz2+3Q2d3maOTPnzMy513F/8N29Zu09e++19qz9OreuzsN7wHvAe8B7wHvAe8B7wHvAe8B7wHugeg+cEPFqX/TTI/KmoX8sIs+rvQeakgd60pk5ER2ajf6BsLzmYUp0raAqXAq3QIvt9sHL3gNN3ANrQvo3FF3nEP1hVVRQuPJPILzvHmLSjuR3Cimj1ag/HA7rQ/KzVLWhMhl8KVRQd4dD4B+wCLgAI1onNOQvyq1LWPZYKib7r4Ma365Qdq6CM+AuaKGAuN4qAnlziC5WdQUlDsEbYks2FFgSvKP3wji+oWjmUj9qnAf3wYPwC6ilsRcsCtpjyL8wzLdhureKYrixYyzyVrgfzodvw5VQ9kuvCTEJFBRTkxS0ZSoNiit5OWxgnG4H+afaBjKSW1DPZKhAUFsb4GBYRCjInT+TpLcVzAmjA/u3kfY2tmknojOufLIT9oFxKBsUcdunuMpd/v0Iit5n4AGnNKlmcXU4S2irNAtqeyQo8rUaKQCLhhMx6C64G2oSWAG1apRiEArNpnvhh6WZx/BzM/o+Kej/06RaHRwUDAqKm+H5cByUDzJHJSuFOqKZWktZLTGXxtyMOQdZM0ZRUY9hmnSuiTHwJfLlE/mmSNDs78b6lgjDpgVlVkXkW3XZlUIzUFrcRwX6ID8I0rT1JXlfW4PhQcFNpKOhnFZU6BLjUbigjIEaA82WwswjSWH+djKWXGxkK54VPGiCzgVJVwoNlj386SZAs/YoeDLMA+2o9E/oZg7Noh51dW7M9uGMMwvmEAWCG++/kW2QyNTW0H2HU6SIQdmVIupd5+C42yft71xnS1MdiEZGNZBC/45pU8ZptTsJ9grYkvR4xIsYrTEot5ocy37ROcJ9Y2uQuxhjJgZ5+uZ6GH2UmGtQ3EurX8Kd0HW4NH04qmdV6HWDtde09TnyQuhun9S2ZhLNFt3g8QJtnX6Fsn9cQY0eiF17AhtlpwJAlw9vBrqfSfU7ThLkGhSuAxqUs+HtcD1Upy1v4jkL3Egltl7Jy+Eb8CO4Hbp8GR61/ySrULgca2T3AdihUJYdbcwIHt02yY2z0tegdgtJUZOgsJ1pyYOuaG3nV/OcxaH+EeqxzribZwt9EEuhK/ONzSyw/EJg82cFttGZNiSw1Y2x0q0w7mbOva+05kHhGtcybjs+yGWkSF8xdepX6zB0RakZ07U9LKxQgXRapXUDJ3snFMiuMFN0oJ4CZeuuIHXjrDHXdioJygZFFrN3VCdeJeM9k9nVyNWKp5sXtSKE4ReU802G9qJ54SIq1k2Ybt2q5ScpOzeA9zsHdcxKWVclr9fa9tPo3AKoLboCQzdsT8GDUNC3rG8u9Va9uWrLETOo+9ag/vYZtKMId9CBOwoKmPog89yoQhnoNTBiGmiw08D9mCWbN6apqMJ3a237c/RPE9wPcCzU9vxBuBBOh22hAmMSnAuVXxXyDoplplc6BKfFBlNBuSDbYsql/ehMVf8Tf0SjAUoDrTTVQlunxvrBrpa298XOOwInaaK1H7xWj2FwMdRvYx2hftidDatC3kGxx/Rqs5GrFdeaF3WojoI+Fod1Tsgh/Z06X8+h3qRV9qdgl6DwzKQvZVSulrZfQp/dmH4V0n9NvgqW0UFekt8qQqo5otJykyfclahWiU8zaGgRdbgVR3tpHbzC0MYo1xi5aKLbOn2HYT8VzThjjx3PbUZvRW2ZHHTorhp5B8U9Qc+mkO6uupcNL2rlmRY8aqm8tiHrKEkzi6Bzx/zDUvH+aOZsrK1Trb25xDR4npGtuNE8rDZyZmKSf/MYSWvr4fdwPCwNsDvRHYLqYLn9P9kVoReltVqo7uXwFGjRgQcFg/JfthkFk7V1ko1i74LZVmpOCxQ7oGxdAZvBUjyOQvn6NuIuP7SVnworQpKgmEWNblCUKpoHw37weSidti4dYdYYQIU7odqYB8+BgoLvayj9IngGLComYpjs1IH3eMDVGKndhmzW+aktdBiDoMO38kbAOOQWFJfR8m9QHSml9BNg1J6frNRQAGof7dpWm9pL/gOfhXlfItBEo2IDrcv2Jxu1F7Vt/Cqa+xbKbo31Wqgzhp61axgKk6BsUKT5cHQL0A0OhF2gAmATXA9147MP5onFVN4T6vzQHbaCy+BKuB8WHaMCA1cV3VBj38fIfeCFsAdsB/XNaUeib+4gzA1Jtk+5Ne4r9h7I2QNlV4rSw3HOffHVew80fQ/4oGj6Y+R7WGMPxJ0pxtCfQSV90u3OuyU6/+g90BQ90IZOPQRLJ39d21cMHV51og/j5Ipr8y94DzSOB3RDGvYNS6cflD28B7wHvAe8B7wHvAe8B7wHvAe8B7wHvAe8B7wH8vbAf7FA4HAlc7+AAAAAAElFTkSuQmCC\" width=\"98.5\" height=\"19\" style=\"width: 98.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e. Your task is to find the total of the sum of the coefficients of the given polynomial function plus the sum of the coefficients of its first derivative plus the sum of cefficients of all its higher degree derivatives.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function totSum = tot_dCoefSum(coef)\r\n  y = x;\r\nend","test_suite":"%%\r\ncs = [5 6 -7 -8];\r\nts = '88';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = [3 15 -2 1];\r\nts = '120';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = [-7 22 43 6 -75 3 1 0 -80 10 5];\r\nts = '-42698751';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = 1:25;\r\nts = '1836856501837772435875025';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = repmat([2,-1],1,15);\r\nts = '47298214022376392514505945712317';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = [ones(1,20) zeros(1,10)];\r\nts = '24893912605687593731774059567276';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = repmat([-2,-25,1],1,10);\r\nts = '-68761759219969440143678420163128';\r\nassert(isequal(tot_dCoefSum(cs),ts))","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":16,"test_suite_updated_at":"2021-08-17T17:53:33.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-16T19:00:56.000Z","updated_at":"2025-11-30T19:39:34.000Z","published_at":"2021-08-17T12:43:32.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider the polynomial function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eP\\\\left(x\\\\right)=5x^3+6x^2-7x-8\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and its first-order derivative \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{dP}{dx}=15x^2+12x-7\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The sums of the coefficients of P and P', are \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e5 + 6 - 7 - 8 = -4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e15+12-7= 20\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, respectively. If we keep summing up coefficients for all higher derivatives the sums sequence will be as follows: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e-4,\\\\ 20,\\\\ 42, ...\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e etc.  The total sum of this sequence converge to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e88\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor this exercise, you are given an array corresponding to the coefficients of a polynomial function. In the example above, the coefficient array is therefore, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e[5\\\\ 6\\\\ -7\\\\ -8]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Your task is to find the total of the sum of the coefficients of the given polynomial function plus the sum of the coefficients of its first derivative plus the sum of cefficients of all its higher degree derivatives.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52532,"title":"Easy Sequences 5: Project Euler Problem 1 - Again!","description":"We are all familiar with Project Euler Problem 1. This time let's try it on bigger multiples and larger range.\r\nFind the sum of all the multiples of the first input or the second input for all positive integers below the third input.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 51px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 25.5px; transform-origin: 407px 25.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72.5px 8px; transform-origin: 72.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWe are all familiar with \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/230\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProject Euler Problem 1\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 181px 8px; transform-origin: 181px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. This time let's try it on bigger multiples and larger range.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 354px 8px; transform-origin: 354px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFind the sum of all the multiples of the first input or the second input for all positive integers below the third input.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function sumMults = euler(mult1,mult2,limit)\r\n    sumMults = mult1 * mult2 / limit;\r\nend","test_suite":"%%\r\nx = '3';\r\ny = '5';\r\nz = '1000';\r\ns = '233168';\r\nassert(isequal(euler(x,y,z),s))\r\n%%\r\nx = '333';\r\ny = '555';\r\nz = '1000000000000000000';\r\ns = '2102102102102102397897897897898053';\r\nassert(isequal(euler(x,y,z),s))\r\n%%\r\nx = '1234567';\r\ny = '67891011';\r\nz = '10000000000000000000000000000';\r\ns = '41236503492327959372976875681892749947271207402890';\r\nassert(isequal(euler(x,y,z),s))\r\n%%\r\nx = '123456789101112';\r\ny = '1234567891011121314151617181920';\r\nz = '10000000000000000000000000000000000000000000';\r\ns = '405000003353802152272811475598036987756535154447670107236880553924296912';\r\nassert(isequal(euler(x,y,z),s))","published":true,"deleted":false,"likes_count":2,"comments_count":1,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":14,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-14T04:00:48.000Z","updated_at":"2025-11-30T19:42:11.000Z","published_at":"2021-08-14T18:07:20.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWe are all familiar with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/230\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProject Euler Problem 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. This time let's try it on bigger multiples and larger range.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind the sum of all the multiples of the first input or the second input for all positive integers below the third input.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52574,"title":"Easy Sequences 8: Triangles with integer sides and prime perimeters","description":"The triangle below is special.\r\n\r\nIt has integer sides and a prime perimeter.  \r\nGiven an integer \"n\" we want to know how many triangles can be formed such that the sides are integers and the perimeters are primes less than or equal to the nearest prime to \"n\". \r\nThe nearest prime number is defined as:\r\n\"n\" if \"n\" itself is prime;\r\neither the previous prime before or the next prime after \"n\", whichever has lesser distance to \"n\"; or\r\nthe previous prime before \"n\" if the previous and next primes are equidistant to \"n\".\r\nAs an example, lets consider \"n = 9\". The nearest prime is 7 because 9 is equidistant between 7 and 11. The primes less than equal to 7 are [2 3 5 7].  There are no integral triangles that can be formed with perimeter 2. For 3 there is one, namely [1 1 1]. For 5 there is one, [1 2 2]. And for 7 there are 2, [1 3 3] and [2 2 3] . So, the total number of prime perimetered integral triangles that can be formed when n = 9 is 4.\r\nNOTE: Rotations and reflections are irrelevant and counted only once.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 451.3px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 225.65px; transform-origin: 407px 225.65px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 91px 8px; transform-origin: 91px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe triangle below is special.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 105px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 52.5px; text-align: left; transform-origin: 384px 52.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: middle;width: 175px;height: 105px\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAArwAAAGiCAYAAAAFjhU1AAAMaGlDQ1BJQ0MgUHJvZmlsZQAASImVlwdYU8kWgOeWJCQkoQQQkBJ6E6RXKSG0AAJSBRshCSSUEBOCir0sKrh2EQEbuiqi6FoAWSv2sij2vlhQUNbFgg2UNyEBXfeV7833nTt/zpw558y5c3PvAKDxkSuR5KKaAOSJC6Tx4cHMsalpTFI7IAFjoAsAQLg8mYQVFxcNGQz2f2/vbwJE0V9zVPj65/h/bdp8gYwHw4yHnMGX8fIgHwcAr+JJpAUAEBV6iykFEgXPgawjhQlCXq3gLCXvUHCGkg8P2CTGsyFfAUCNyuVKswCg34d6ZiEvC/qh90F2FvNFYgA0RkAO4Am5fMiK3Efk5eUruByyLbSXQIb5AO+M73xm/c1/xpB/LjdriJXrGmhqISKZJJc77f8szf9uebnywRjWUKhCaUS8Yv2whrdz8qMUTIXcJc6IiVXUGvJHEV9ZdwBQilAekaS0R414MjasH9CD7MznhkRBNoIcJs6NiVbpMzJFYRzIcLegU0UFnETI+pAXCWShCSqbTdL8eFUstD5Tymap9Oe40oG4ilgP5TlJLJX/N0IBR+UfoxcJE1MgUyBbFoqSYyDTITvJchKiVDajioTsmEEbqTxekb8l5HiBODxY6R8rzJSGxavsS/Jkg+vFNglFnBgV7ysQJkYo64Od4nEH8odrwa4IxKykQT8C2djowbXwBSGhyrVjHQJxUoLKz0dJQXC8ci5OkeTGqexxc0FuuEJvDtldVpigmosnF8DNqfSPZ0oK4hKVeeJF2dzIOGU++HIQDdggBDCBHEoGyAfZQNTS1dAFfylHwgAXSEEWEABHlWZwRsrAiBheE0AR+BOSAMiG5gUPjApAIdR/GdIqr44gc2C0cGBGDngGOQ9EgVz4Wz4wSzwULRk8hRrRP6JzofBgvrlQFOP/Xj+o/aZhQU20SiMfjMjUGLQkhhJDiBHEMKIdbogH4H54NLwGQXHFvXGfwXV8syc8I7QSHhNuENoIdyaJ5kl/yHI0aIP+w1S1yPi+Frg19OmBB+P+0Dv0jOvhhsARd4dxWHggjOwBtWxV3oqqMH/w/bcVfHc3VHZkZzJKHkYOItv+OJNuT/cY8qKo9ff1UeaaMVRv9tDIj/HZ31WfD/uoHy2xRdh+7Cx2AjuPHcYaABM7hjVil7AjCh7aXU8HdtdgtPiBfHKgH9E/4nFVMRWVlDnXOnc69ynHCgRTCxQPHjtfMk0qyhIWMFnw7SBgcsQ8pxFMV2dXFwAU7xrl31f35YF3CGKg9U03xwGAUYT+/v5D33QxJQDUz4aP/5tvOlvY02YAcG4uTy4tVOpwxYUA/yU04JNmAEyABbCF63EFnsAPBIFQEAliQSJIBRNhlYVwn0vBFDADzAXFoBQsB2tABdgItoAdYDfYBxrAYXACnAEXwRVwA9yDu6cdvATd4D3oRRCEhNAQBmKAmCJWiAPiingjAUgoEo3EI6lIOpKFiBE5MgOZj5QiK5EKZDNSg/yKHEJOIOeRVuQO8gjpRN4gn1EMpaI6qDFqjY5EvVEWGoUmohPQLHQyWoQuQJei5Wg1ugutR0+gF9EbaBv6Eu3BAKaO6WFmmCPmjbGxWCwNy8Sk2CysBCvDqrE6rAne52tYG9aFfcKJOANn4o5wB0fgSTgPn4zPwpfgFfgOvB4/hV/DH+Hd+FcCjWBEcCD4EjiEsYQswhRCMaGMsI1wkHAaPkvthPdEIlGPaEP0gs9iKjGbOJ24hLieuId4nNhKfELsIZFIBiQHkj8plsQlFZCKSetIu0jHSFdJ7aSPaupqpmquamFqaWpitXlqZWo71Y6qXVV7rtZL1iRbkX3JsWQ+eRp5GXkruYl8mdxO7qVoUWwo/pRESjZlLqWcUkc5TblPeauurm6u7qM+Rl2kPke9XH2v+jn1R+qfqNpUeyqbOp4qpy6lbqcep96hvqXRaNa0IFoarYC2lFZDO0l7SPtIZ9Cd6Bw6nz6bXkmvp1+lv9Iga1hpsDQmahRplGns17is0aVJ1rTWZGtyNWdpVmoe0ryl2aPF0HLRitXK01qitVPrvFaHNknbWjtUm6+9QHuL9kntJwyMYcFgM3iM+YytjNOMdh2ijo0ORydbp1Rnt06LTreutq67brLuVN1K3SO6bXqYnrUeRy9Xb5nePr2bep+HGQ9jDRMMWzysbtjVYR/0h+sH6Qv0S/T36N/Q/2zANAg1yDFYYdBg8MAQN7Q3HGM4xXCD4WnDruE6w/2G84aXDN83/K4RamRvFG803WiL0SWjHmMT43BjifE645PGXSZ6JkEm2SarTY6adJoyTANMRaarTY+ZvmDqMlnMXGY58xSz28zILMJMbrbZrMWs19zGPMl8nvke8wcWFAtvi0yL1RbNFt2WppajLWdY1lretSJbeVsJrdZanbX6YG1jnWK90LrBusNG34ZjU2RTa3PflmYbaDvZttr2uh3Rztsux2693RV71N7DXmhfaX/ZAXXwdBA5rHdoHUEY4TNCPKJ6xC1HqiPLsdCx1vGRk55TtNM8pwanVyMtR6aNXDHy7Mivzh7Ouc5bne+5aLtEusxzaXJ542rvynOtdL3uRnMLc5vt1uj22t3BXeC+wf22B8NjtMdCj2aPL55enlLPOs9OL0uvdK8qr1veOt5x3ku8z/kQfIJ9Zvsc9vnk6+lb4LvP9y8/R78cv51+HaNsRglGbR31xN/cn+u/2b8tgBmQHrApoC3QLJAbWB34OMgiiB+0Leg5y46VzdrFehXsHCwNPhj8ge3Lnsk+HoKFhIeUhLSEaocmhVaEPgwzD8sKqw3rDvcInx5+PIIQERWxIuIWx5jD49RwuiO9ImdGnoqiRiVEVUQ9jraPlkY3jUZHR45eNfp+jFWMOKYhFsRyYlfFPoiziZsc99sY4pi4MZVjnsW7xM+IP5vASJiUsDPhfWJw4rLEe0m2SfKk5mSN5PHJNckfUkJSVqa0jR05dubYi6mGqaLUxjRSWnLatrSecaHj1oxrH+8xvnj8zQk2E6ZOOD/RcGLuxCOTNCZxJ+1PJ6SnpO9M7+PGcqu5PRmcjKqMbh6bt5b3kh/EX83vFPgLVgqeZ/pnrszsyPLPWpXVKQwUlgm7RGxRheh1dkT2xuwPObE523P6c1Ny9+Sp5aXnHRJri3PEp/JN8qfmt0ocJMWStsm+k9dM7pZGSbfJENkEWWOBDvyovyS3lf8kf1QYUFhZ+HFK8pT9U7WmiqdemmY/bfG050VhRb9Mx6fzpjfPMJsxd8ajmayZm2chszJmNc+2mL1gdvuc8Dk75lLm5sz9fZ7zvJXz3s1Pmd+0wHjBnAVPfgr/qbaYXiwtvrXQb+HGRfgi0aKWxW6L1y3+WsIvuVDqXFpW2reEt+TCzy4/l//cvzRzacsyz2UblhOXi5ffXBG4YsdKrZVFK5+sGr2qfjVzdcnqd2smrTlf5l62cS1lrXxtW3l0eeM6y3XL1/VVCCtuVAZX7qkyqlpc9WE9f/3VDUEb6jYabyzd+HmTaNPtzeGb66utq8u2ELcUbnm2NXnr2V+8f6nZZritdNuX7eLtbTvid5yq8aqp2Wm0c1ktWiuv7dw1fteV3SG7G+sc6zbv0dtTuhfsle998Wv6rzf3Re1r3u+9v+6A1YGqg4yDJfVI/bT67gZhQ1tjamProchDzU1+TQd/c/pt+2Gzw5VHdI8sO0o5uuBo/7GiYz3HJce7TmSdeNI8qfneybEnr58ac6rldNTpc2fCzpw8yzp77Jz/ucPnfc8fuuB9oeGi58X6Sx6XDv7u8fvBFs+W+stelxuv+Fxpah3VevRq4NUT10KunbnOuX7xRsyN1ptJN2/fGn+r7Tb/dsed3Duv7xbe7b035z7hfskDzQdlD40eVv9h98eeNs+2I49CHl16nPD43hPek5dPZU/72hc8oz0re276vKbDteNwZ1jnlRfjXrS/lLzs7Sr+U+vPqle2rw78FfTXpe6x3e2vpa/73yx5a/B2+zv3d809cT0P3+e97/1Q8tHg445P3p/Ofk75/Lx3Sh+pr/yL3Zemr1Ff7/fn9fdLuFLuwKcABgXNzATgzXb4nZAKAAOe2yjjlGfBgYYoz68DBP4TK8+LA80TgDrYKT7j2ccB2AvFGgoVSmwQAIlBAHVzGxJVk2W6uSp90WsBIJn197/JB4AMpS+8v783rr//SxVM9joARzuUZ1BFI8IzwyYXBV01PVAFfmjK8+l3a/yxB4oM3MGP/b8AVM6O+r8W+fwAAACKZVhJZk1NACoAAAAIAAQBGgAFAAAAAQAAAD4BGwAFAAAAAQAAAEYBKAADAAAAAQACAACHaQAEAAAAAQAAAE4AAAAAAAAAkAAAAAEAAACQAAAAAQADkoYABwAAABIAAAB4oAIABAAAAAEAAAK8oAMABAAAAAEAAAGiAAAAAEFTQ0lJAAAAU2NyZWVuc2hvdMms6Q4AAAAJcEhZcwAAFiUAABYlAUlSJPAAAAHWaVRYdFhNTDpjb20uYWRvYmUueG1wAAAAAAA8eDp4bXBtZXRhIHhtbG5zOng9ImFkb2JlOm5zOm1ldGEvIiB4OnhtcHRrPSJYTVAgQ29yZSA2LjAuMCI+CiAgIDxyZGY6UkRGIHhtbG5zOnJkZj0iaHR0cDovL3d3dy53My5vcmcvMTk5OS8wMi8yMi1yZGYtc3ludGF4LW5zIyI+CiAgICAgIDxyZGY6RGVzY3JpcHRpb24gcmRmOmFib3V0PSIiCiAgICAgICAgICAgIHhtbG5zOmV4aWY9Imh0dHA6Ly9ucy5hZG9iZS5jb20vZXhpZi8xLjAvIj4KICAgICAgICAgPGV4aWY6UGl4ZWxZRGltZW5zaW9uPjQxODwvZXhpZjpQaXhlbFlEaW1lbnNpb24+CiAgICAgICAgIDxleGlmOlBpeGVsWERpbWVuc2lvbj43MDA8L2V4aWY6UGl4ZWxYRGltZW5zaW9uPgogICAgICAgICA8ZXhpZjpVc2VyQ29tbWVudD5TY3JlZW5zaG90PC9leGlmOlVzZXJDb21tZW50PgogICAgICA8L3JkZjpEZXNjcmlwdGlvbj4KICAgPC9yZGY6UkRGPgo8L3g6eG1wbWV0YT4KlbbGDQAAABxpRE9UAAAAAgAAAAAAAADRAAAAKAAAANEAAADRAAA68GPNZMEAADq8SURBVHgB7N0HmNTU+j/wd3tlGyBFEeEHijSl9957XfrSpEmRC0rvIFxgBQEREOlVWDpL7yBF4Ar/q0gVEQEvXNyF7X3/eeOd2WQy22d2ksw3z3OfSU6Sk3M+WS/vnDnFIVXYCBsEIAABCEAAAhCAAAR0KuCAgFenbxbVggAEIAABCEAAAhAQBRDw4g8BAhCAAAQgAAEIQEDXAgh4df16UTkIQAACEIAABCAAAQS8+BuAAAQgAAEIQAACENC1AAJeXb9eVA4CEIAABCAAAQhAAAEv/gYgAAEIQAACEIAABHQtgIBX168XlYMABCAAAQhAAAIQQMCLvwEIQAACEIAABCAAAV0LIODV9etF5SAAAQhAAAIQgAAEEPDibwACEIAABCAAAQhAQNcCCHh1/XpROQhAAAIQgAAEIAABBLz4G4AABCAAAQhAAAIQ0LUAAl5dv15UDgIQsCeB1NRURXUdHBwUaUiAAAQgYG8CCHjt7Y2jvhCAgC4FzAW7XFEEvLp83agUBCCQTQEEvNkEw+UQgAAE1CiAgFeNbwVlggAE1CKAgFctbwLlgAAEIJBDAQS7OYTDbRCAgN0IIOC1m1eNikIAAnoVQMCr1zeLekEAApYSQMBrKUnkAwEIQMAGAq9fvyZfX18yDXrRd9cGLwOPhAAEVCuAgFe1rwYFgwAEIJCxwIMHD6hv37505swZcnV1lV2MgFfGgQMIQMDOBRDw2vkfAKoPAQhoVyAwMJCuXbtGFStWpA0bNlBAQICxpRcBr3bfK0oOAQhYXgABr+VNkSMEIAABqwu8fPmSqlevbpx2LH/+/BQaGkpvvPGG1Z+NB0AAAhDQmgACXq29MZQXAhCAgCCwePFiWrJkCbm4uIgeKSkp5OPjQ1u2bKEKFSrACAIQgAAEJAIIeCUY2IUABCCgFQF/f3+xC4NpeTl97969VKxYMdNTOIYABCBgtwIIeO321aPiEICAVgVu3bpFLVq0IDc3N7NVcHJyopCQEPrggw/MnkciBCAAAXsTQMBrb28c9YUABDQvMG7cONq9e7ex/66hQjw1mWGwGndv4GtKlixpOI1PCEAAAnYrgIDXbl89Kg4BCGhRIDExkWrVqkXh4eGZFp/79Z49e5aKFy+e6bW4AAIQgICeBRDw6vntom4QgIDuBLirwpgxYxTdGZKSksjZ2VlRX29vbzp9+jTxLA7YIAABCNirAAJee33zqDcEIKBJgdKlSxO33Eo3bvXt1asX7dy509ilQXre3d2dTp06RYULF5YmYx8CEICA3Qgg4LWbV42KQgACWhc4dOgQjRo1ShbUcvDbrFkzWr16tXhu//79xqnKpPXlAW43btxQtAxLr8E+BCAAAb0KIODV65tFvSAAAd0JDBo0SFxG2LRiHOSWL19eTJ40aRJ999135OjoKLuMB7QVKlSIjh49Sr6+vrJzOIAABCCgdwEEvHp/w6gfBCCgC4Ho6Gji7gweHh6y+vB8uxzEcrcFwxYUFEQXLlwgnp5MunHQ6+XlJS5HLL1eeg32IQABCOhRAAGvHt8q6gQBCOhO4KOPPqKTJ0/KBqbFx8fTtGnTaOjQoYr6fvLJJ+JSw4ZpygwXcNDLQfKZM2cUrcCGa/AJAQhAQG8CCHj19kZRHwhAQHcCL168oNq1axMHq9KNuy3cv39fmiTb79u3L507d04WJPMFycnJYtB75MgR4lkcsEEAAhDQuwACXr2/YdQPAhDQvMC2bdvEllxpRTj45enJeBBbehsHtv3796dLly6ZvaRUqVJ07Ngxs+eQCAEIQEBPAgh49fQ2URcIQECXAg0aNKAnT57I6ubq6kqHDx+mEiVKyNJND3gWh2HDhon9fF1cXGSneTqzChUq0K5duxR9g2UX4gACEICAxgUQ8Gr8BaL4EICAvgW+//576tq1qzjYzFBTbt3l1tnjx48bkjL85KA3MDCQfvzxR7P9dmvWrEmbN282ey7DjHESAhCAgEYEEPBq5EWhmBCAgH0KNGnShH777TfZ3LsRERHiALYPPvggyygc9Pbp04cuX76sCGx5lTae1mzfvn1m5/DN8kNwIQQgAAGVCiDgVemLQbEgAAEI8IC0du3aEXc9kG456XvLrcIJCQnUuXNn+vnnnxUD2fh8o0aNaM2aNbLgWvpc7EMAAhDQqgACXq2+OZQbAhDQvcDChQtp1apVigB03rx51L179xzXn1dme/jwoeJ+DnrLli1LvJCF6Ry+iouRAAEIQEBDAgh4NfSyUFQIQMC+BEqWLKkIdnle3evXr5Ofn1+OMaKioqh169b0+PFjRWDLXR+4GwW39GKDAAQgoBcBBLx6eZOoBwQgoCuBJUuW0KJFi4hnYzBs3CWBlxeeMWOGISlXn+Zmf+AMeTqzevXq0caNGxX9fXP1QNwMAQhAwEYCCHhtBI/HQgACEEhPIDY2llq0aEFPnz6VXcKB6IMHDxT9b2UXZeMgMjJSDGx5EJzpimxo6c0GJC6FAARUL4CAV/WvCAWEAATsTeCnn36iDh06KIJQHsDGLb+W3HhAXOPGjenZs2eKbPkcD3Kz9DMVD0ICBCAAASsLIOC1MjCyhwAEIJBdAZ4+7OLFi7LuBNziyl0M6tevn93sMr2eW3irVKlC/AzTjVuVO3XqRF9++aXpKRxDAAIQ0IwAAl7NvCoUFAIQsAcBnnO3Ro0alC9fPmN1efaEAgUK0NWrV41plt7566+/qFWrVsSfphu39Pbr14/mzJljegrHEIAABDQhgIBXE68JhYQABOxFYPTo0XTw4EFZd4b4+Hhav3498XRi1tz+85//UMOGDcX5ek379HK/4rFjx9Jnn31mzSIgbwhAAAJWEUDAaxVWZAoBCEAg+wIcVHKXhbCwMNnNRYoUoXPnzimmEJNdZKEDHijHU5ZFR0cTtyxLN+7yMGrUKDHwlaZjHwIQgIDaBRDwqv0NoXwQgIDdCBw5ckScdszd3d1YZw46g4KCaNasWcY0a+/cu3ePunbtSjxfr2lLL8/sMHPmTBo+fLi1i4H8IQABCFhMAAGvxSiREQQgAIHcCbz77rviHLjSXOLi4ujy5cv09ttvS5Otvv/o0SNq06YN8fPNbRz0ciCODQIQgIAWBBDwauEtoYwQgIDuBQ4fPkwff/yxbI5dniGB+9Ru2LDBJvW/desWdenShXjQmunGgfCUKVPQ0msKg2MIQECVAgh4VflaUCgIQMDeBHgFtTNnzsiqzQHv+fPn87x1V1oInhO4W7du4kA2aTrvc3cLXg2Opy3DBgEIQEDNAgh41fx2UDYIQMAuBLi1tEyZMopBaaVKlaLQ0FBycXGxqQP36W3evLmifFwoDsrHjx+Pll6bviE8HAIQyEwAAW9mQjgPAQhAwMoCEyZMoO3bt8sCW+5GMGnSJLGbg5Ufn6Xsv//+e+rbt69iEBvfzC29vDAFrw6HDQIQgIAaBRDwqvGtoEwQgIDdCISHh1PlypVlq6px5Tngffz4saocbt++LS5O4eTkpChXUlISzZ8/n3r27Kk4hwQIQAACthZAwGvrN4DnQwACdi3AywXPnj1bZsDdBLh1d+jQobJ0NRzwfMDcp9fb21tRHJ7CbOnSpeLsDoqTSIAABCBgQwEEvDbEx6MhAAEItG3blrjlVLp5eHjQ8ePHqWjRotJk1exfuXKFevfubbY83B+Zg/imTZuaPY9ECEAAArYQQMBrC3U8EwIQgIAgcPPmTbGLgKenp9GD+8NWqFCB9u/fb0xT486xY8eoT58+5OPjoyge12HHjh1UrVo1xTkkQAACELCFAAJeW6jjmRCAAAQEAV7Cl1t3HR0djR68khnPyVu1alVjmlp3eAaJ0aNHK4rHXRu4pTckJIRq1aqlOI8ECEAAAnktgIA3r8XxPAhAAAKCwJ07d8RZDXiwl3QrXbo0HT16VJqk6v21a9fSjBkzyM3NTVFODnp5bmFeQQ4bBCAAAVsKIOC1pT6eDQEI2K0AT+O1fPlyRf150Bf369XStm3bNnHVNWlLtaH8nLZnzx6xm4YhDZ8QgAAE8loAAW9ei+N5EIAABAQBbslNSUmRWXBf3rNnz1L+/Pll6Vo4WLx4MS1btszs4hQJCQniinElSpTQQlVQRghAQIcCCHh1+FJRJQhAQN0C3377Lc2cOZPc3d2NBeV5d7t3707BwcHGNK3trF+/nubMmWN2cQoO5rlvcrFixbRWLZQXAhDQgQACXh28RFQBAhDQjgAHtnXq1KG//vpLVmhuBeUBbObmt5VdqPKDhQsX0ldffUWurq6ykvLMDc7OzmLQy0smY4MABCCQlwIIePNSG8+CAATsXoCnIuvatau4HK8Uo02bNmKXAGmaVvfT65/M9cmXL584kM3f31+r1UO5IQABDQog4NXgS0ORIQAB7QoMGzZMXFSCp+4ybLy/bt06ql+/viFJ859Lliwhbu3lRTSkG/db9vX1FQeylSxZUnoK+xCAAASsJoCA12q0yBgCEICAXIC7MfAUXX5+frIT3L/1p59+kqXp4WD69Om0adMmswPZuIWXV5MLCAjQQ1VRBwhAQOUCCHhV/oJQPAhAQD8CI0aMEPuwSqfvio2NJZ6KLDAwUD8VldSEuzcsWrRIMU8v9+nl7g0c9BYqVEhyB3YhAAEIWF4AAa/lTZEjBCAAAYUAr6DWpEkTxWA1/nn/xx9/VFyvp4SJEyfS5s2bZbNSGOrHszbwMsrsgA0CEICAtQQQ8FpLFvlCAAIQkAicPHmSBg4cSC4uLpJUov79+9O0adNkaXo8+PTTT8V+u9LWbUM9uZ/vpUuXyMfHx5CETwhAAAIWFUDAa1FOZAYBCEDAvEDVqlUpLCxMNkctT0V26tQpu1l6d8iQIWKXDnPLEBcvXpxOnDhhtr+veVGkQgACEMi6AALerFvhSghAAAI5EuCgtl+/frJ+rDxbQdmyZenQoUM5ylOrN3GL9vnz52WBP9eF+/S+8cYb4hcALy8vrVYP5YYABFQqgIBXpS8GxYIABPQhwIEtB3kXL16UVSguLk5Me+edd2Tpej9gjy5duoj9lnkhCunGQS+39J45c0aajH0IQAACuRZAwJtrQmQAAQhAIH0BHqxWvXp14u4L0q1MmTJ04MABu/0JPygoSAz4pfMRs09ycjK99957dPDgQcVqbVI/7EMAAhDIjgAC3uxo4VoIQAAC2RRYsGABff3117LBahzUjR8/noYPH57N3PR1Oc9a8fDhQzIdyMYtvbwoBQ/0wwYBCEDAEgIIeC2hiDwgAAEImBHgOXa5n65pQBcfH0+///673bbuGqg4sO3VqxddvXrVkGT85C8FVapUod27d4v9e01bgo0XYgcCEIBAFgQQ8GYBCZdAAAIQyIkALxc8Z84cWcCblJREI0eOpHHjxuUkS93dw316mzZtKrb0Ojk5yerHVrVq1aLt27crBrnJLsQBBCAAgUwEEPBmAoTTEIAABHIq0LFjR8WSwTwl17lz56hgwYI5zVZ39/EAvq5du9Lt27cVdeOAuFGjRsRfHrBBAAIQyKkAAt6cyuE+CEAAAhkI/Pbbb1SnTh3iRRWkW+XKlWnHjh2yVl/peXvdj46Ops6dO9Pdu3cVXT04IOYvD8uXL0dLr73+gaDeEMilAALeXALidghAAALmBDhAu3nzpix44z69mzZtEn/CN3ePvadxYNuqVSt6/PixWYp27drRkiVLzJ5DIgQgAIGMBBDwZqSDcxCAAARyIPDgwQNq1qyZrBWXB2gVKVJEMR9vDrLX9S0RERHUoUMHevTokcyPK82D/Tgg/vbbb3VtgMpBAAKWF0DAa3lT5AgBCNi5wKJFi2jFihUyBe6LumzZMuJWSmzpC/AXg6ioKGrZsiU9e/ZMEfTyeZ7Dd9asWelngjMQgAAETAQQ8JqA4BACEIBAbgUqVqxI3CdVuvn6+tKJEycof/780mTspyPA3Rtq165Nr1+/VlzBU5a1aNGCvvnmG8U5JEAAAhAwJ4CA15wK0iAAAQjkUGDnzp00evRo8vT0NObAARq3WK5atcqYhp3MBZ4/fy52YQgPD1e09HKL+aBBg2jKlCmZZ4QrIAABuxdAwGv3fwIAgAAELClgrnWXlxe+c+cOBQQEWPJRdpEXL8lcqVIl4hZf043n6e3ZsyfNnz/f9BSOIQABCMgEEPDKOHAAAQhAIOcC165dE1cO49ZHw8Z9TnkAG35+N4hk/5NnbeC5eM1tbD106FCaOHGiudNIgwAEICAKIODFHwIEIAABCwmMGTOG9u/fL5srlpfE3bx5s7himIUeY5fZvHz5kho3bqzoG80Y3PrL9p999pld2qDSEIBA5gIIeDM3whUQgAAEMhXgOXZLlChBXl5esmt54Yl///vfij6osotwkCUBnqqMF/Nwd3dXXM/9pCdMmEAff/yx4hwSIAABCCDgxd8ABCAAAQsIcOtiSEgIOTs7G3PjlseZM2fS4MGDjWnYyZ3Aw4cPxXl6Y2JiFBlx2ty5c2ngwIGKc0iAAATsWwABr32/f9QeAhCwgMCrV6/E1dN4NgHpxsEvL5WLzbICP/30E7Vp04ZcXFwUGXNLO8+G0b59e8U5JEAAAvYrgIDXft89ag4BCFhI4OTJkzRkyBBZ313Omlt2MZjKQsgm2XA3kcDAQOKZGkw3TgsODqZu3bqZnsIxBCBgpwIIeO30xaPaEICA5QR4EYT79+/LAl6eneHAgQNUvnx5yz0IOckEeCEP7rPL1qYbr9a2detWseXd9ByOIQAB+xNAwGt/7xw1hgAELChw+fJl6ty5s2yhCZ4qq3jx4nT27FkLPglZmRO4fv26OBevdCo4w3WOjo7iEs88LRw2CEDAvgUQ8Nr3+0ftIQCBXAhwyyIvfHD16lVZ6y4PnuJuDuXKlctF7rg1qwLHjh2jYcOGmZ0JgxeuWLlyJbVt2zar2eE6CEBAhwIIeHX4UlElCEAgbwR4blheEMF0xoD33nuPDh8+nDeFwFNEgfPnz4tLDfP0ZKYbz4XM3Rtq1KhhegrHEICAnQgg4LWTF41qQgAClhfg1dN4GizpbAHc6jtu3DjMB2t57kxzPH36NAUFBZmdp5dvXrZsmTi7Q6YZ4QIIQEB3Agh4dfdKUSEIQCCvBN566y1yc3OTPS4xMVFcaMLPz0+WjoO8Edi7dy+NHTvWbPcGJycn2rFjB1WqVClvCoOnQAACqhFAwKuaV4GCQAACWhJYv369uKiEdKEJng6LWxg///xzLVVFd2U9deoU9enTRzaQ0FBJboHfuHEj1atXz5CETwhAwA4EEPDawUtGFSEAAcsKcD/RLl26EC+AIN14VoBr164RWnelKrbZ37Jli9i1xNPTU1EAXpqYV8UrW7as4hwSIAABfQog4NXne0WtIAABKwo8f/6cqlSpoujOUL16dXFwFAe+2GwvwN0Xxo8fL1vu2VAqXvaZ+/zyAENsEICA/gUQ8Or/HaOGEICAhQV69epFly5dIu4Tath4+qvly5dThw4dDEn4VIEADyycMWMGeXl5KUrDLb1Hjhyht99+W3EOCRCAgL4EEPDq632iNhCAgJUFHj9+TLVr15a17nK/UA6efvnlFys/HdnnRGDp0qXE/+PpyUw3HmTIC4SULFnS9BSOIQABHQkg4NXRy0RVIAAB6wvMnz+fvv32W9mDOGhatWoVtW7dWpaOA/UIzJ49m1avXi37omIoHU8rx32v8+XLZ0jCJwQgoDMBBLw6e6GoDgQgYF0Bbt3lPrzSjQepcX9QX19faTL2VSbAX1a4i4NpH2tuoefBbcePH6eiRYuqrNQoDgQgYAkBBLyWUEQeEICAXQhwQNS/f3/y8PAw1peDpQYNGhBPU4ZN/QIjR46k0NBQWf9rQ6m5W8q//vWvdBeuMFyHTwhAQHsCCHi1985QYghAwEYCVatWpbCwMFlf0FevXokLTbz55ps2KhUem10Bnid57dq1Zlt633jjDTp27Bha67OLiushoHIBBLwqf0EoHgQgoA6BK1euiIsZcIuuYeP9WrVqiVORGdLwqQ2BESNG0MGDB2XLQnPJ+Z0WLFiQTpw4QT4+PtqoDEoJAQhkKoCAN1MiXAABCECAxPlcd+/eraDguV655Reb9gQmT54sLjVsWnIOerkvL6/YZrp0tOm1OIYABLQhgIBXG+8JpYQABGwowEsGly5dWvETuL+/P124cEHWp9eGxcSjcyAwevRo2rVrl6LfLq+m984779C+ffuwcl4OXHELBNQmgIBXbW8E5YEABFQnMHPmTFq3bp3s529eaGLMmDE0duxY1ZUXBcqewNChQ8UuDObm6eUvOvv370dLb/ZIcTUEVCeAgFd1rwQFggAE1CQQERFBdevWpejoaEWxfv31V0UaErQnwF0YePaGAwcOkKurq6wC3LpfqlQpOnz4MIJemQwOIKAtAQS82npfKC0EIJDHAjxif/jw4YqnDhgwgKZOnapIR4L2BDjg5dZdXjL64sWL5OzsrKhE5cqVadu2bbJWfsVFSIAABFQrgIBXta8GBYMABNQg0KlTJ3HaMWlZnJyciAewVahQQZqMfR0IdOvWTVx1zXRxipSUFHr77bfpzJkzir7cOqg2qgAB3Qsg4NX9K0YFIQCBnArcvn2b6tevL5ueilsDixcvLgY+Oc0X96lTgN8tB7ZdunShGzduKFp6+TyvtLdlyxZ1VgClggAE0hVAwJsuDU5AAAL2LtC5c2e6efOmbKGJmJgY2rlzJ9WrV8/eeXRd/+bNm5O5PtocEL/33nt06NAhs6u16RoFlYOAhgUQ8Gr45aHoEICA9QSeP39OzZo1UwxW49bd06dPW+/ByFkVArGxsdSiRQv6448/FF0YuKW3Zs2aYp9eVRQWhYAABDIVQMCbKREugAAE7FFg06ZNNG3aNMXP2hMmTKAhQ4bYI4ld1pm/9HBLr+mUZYmJiWJ3l82bNyvO2SUUKg0BlQsg4FX5C0LxIAAB2wiULFlSEcjwz9mXL1+mwoUL26ZQeGqeC/B0dLx8dFRUlOLvgVt6GzVqRGvXrs3zcuGBEIBA9gQQ8GbPC1dDAAJ2IMCtdpMmTZLNu8rzsbZq1YpWrVplBwKoolSAuze0bduWHj16JE0W93kBko4dO9JXX32lOIcECEBAPQIIeNXzLlASCEBABQIc2Hbo0IHu3LkjKw0HNrdu3ZLN2CC7AAe6FoiMjKQ6deqYbenlv43BgwfT9OnTdW2AykFAywIIeLX89lB2CEDA4gJ//vmnuLKaacY8SGnr1q2myTi2I4FXr15R+/bt6enTp4pac5/eoKAgmjt3ruIcEiAAAdsLIOC1/TtACSAAARUJjBo1ig4ePCibcio5OZmCg4MpMDBQRSVFUWwhwF+IuHtDWFiYYvYGnrJu7NixNG7cOFsUDc+EAAQyEEDAmwEOTkEAAvYl8OLFC6pUqRK5u7vLKs6Dkx4+fChLw4H9CnCwy7M3cIuvuY1n8eDZPLBBAALqEUDAq553gZJAAAI2Fpg3bx6tWbNGNho/Pj6evvjiC+rRo4eNS4fHq0mApyrjwWo8i4PplGVxcXH0ySefIOhV0wtDWexeAAGv3f8JAAACEGABHqzGrXaPHz+Wgfj4+NDFixfJ09NTlo4DCHD3hpYtW4oD2Uw1+FeByZMn06BBg0xP4RgCELCBAAJeG6DjkRCAgPoEOKjt3r27bCoyDlp4iVlMRaa+96WWEj158oQaNmxI/LdiuvGXqGHDholT3JmewzEEIJC3Agh489YbT4MABFQqUK9ePeLgxdHR0VhC/rn65MmTVL58eWMadiBgKnD//n1xKjvuymDavYEXK+GZG3r16mV6G44hAIE8FEDAm4fYeBQEIKBOgStXroh9dJ2cnIwF5Ba7ihUr0r59+4xp2IFAegLPnj0TV2RzdnZWXMItvePHj6cRI0YoziEBAhDIGwEEvHnjjKdAAAIqFuBppPbs2SMrIbfM7dy5k6pVqyZLxwEE0hO4ceOG2NLr4uKiuIS/QHFLb8+ePRXnkAABCFhfAAGv9Y3xBAhAQMUCHIi8++67xAGudCtSpAgdO3aMvLy8pMnYh0CGArxCX+vWrRVdG/gmnqeXlyDu2rVrhnngJAQgYHkBBLyWN0WOEICAhgQWL15M/D83NzdjqfknaF4qdurUqcY07EAgqwKXLl2iTp06kbe3t9lbvvnmG2ratKnZc0iEAASsI4CA1zquyBUCENCAAM+xW65cOcUI+4SEBOJ5Vl1dXTVQCxRRjQIXLlyggQMHKn454LJGRkbStm3bEPSq8cWhTLoVQMCr21eLikEAApkJHDlyhHgpYemUUrzfp08fmj17dma34zwEMhTgvy+eh9d05T6+iX9F4CWseWAkNghAwPoCCHitb4wnQAACKhXgqaJ++OEHWel4lP3+/fupTJkysnQcQCAnAkePHqWRI0fKvlQZ8uGgl1t669SpY0jCJwQgYCUBBLxWgkW2EICAugWeP39OZcuWJT8/P1lBixYtSvxzNDYIWEpgw4YNNGvWLNkcz4a8ExMT6fjx4/iCZQDBJwSsJICA10qwyBYCEFC3AK+qdvXqVVkQwqPo165dK46yV3fpUTqtCRw6dEjsPmO6MAXXg39V2L59O1WuXFlr1UJ5IaAZAQS8mnlVKCgEIGApAV4koFmzZsQrY0k3f39/un79ujQJ+xCwmAAvUT1v3jxKb57eAwcOYFU/i2kjIwjIBRDwyj1wBAEI2IEAt6ZNmTJFMVfq9OnTqV+/fnYggCraSmDHjh00efJks4/nwW08kK1kyZJmzyMRAhDIuQAC3pzb4U4IQECjApUqVaKIiAhZ6R0dHenMmTP01ltvydJxAAFLC6xbt078wuXp6SnLmmcI8fDwELs3YPYGGQ0OIJBrAQS8uSZEBhCAgJYE9u7dSyNGjBADC0O5k5OTxSWEQ0JCDEn4hIBVBbh7w8KFCxW/MvBDeXW/w4cP48uXVd8AMrc3AQS89vbGUV8I2LEAB7atWrUSF5WQMkRFRdHt27cpICBAmox9CFhVgFt6uXuD6fLVhpZensf37bfftmoZkDkE7EUAAa+9vGnUEwIQoEePHlGLFi3ESf+lHDVr1qStW7dKk7APgTwRWLBgAS1fvlyxqh/P5sBT5oWGhlLhwoXzpCx4CAT0LICAV89vF3WDAARkAlOnThUDW+6va9i4NS04OJi6dOliSMInBPJUYP78+bRixQqzszdwQXhxlAIFCuRpmfAwCOhNAAGv3t4o6gMBCJgV4G4LpUuXJtOBQnzxvXv3yMnJyex9SIRAXghMnDiRtmzZQm5uborHFSpUiE6cOKHo+qC4EAkQgEC6Agh406XBCQhAQE8CPP/p6tWrZYFtQkICTZo0iYYPH66nqqIuGhXgJYh5sJrp4hT8KwQHwpcvX1asDKjRqqLYEMhzAQS8eU6OB0IAAnktwAtM8GC1x48fyx7Nrb280IS5VjXZhTiAQB4J8Jcv7rdrbnEK7st78eLFPCoJHgMBfQkg4NXX+0RtIAABMwI3b96kdu3aKQYGtW7dmpYtW6ZoUTOTBZIgkGcCPG0et/RK+5rzw1NSUsQBbMePH6d8+fLlWXnwIAjoQQABrx7eIuoAAQhkKMDLCD948EAWQHB3Bp53l2dowAYBtQkEBgaKvz6YBr1cTl4c5dy5c2orMsoDAVULIOBV9etB4SAAgdwKcOtumzZtiJdtNWzcJ5LnNz179qwhCZ8QUJ0Ad284evSo4heIpKQkKlOmDB04cED81cK0z6/qKoICQUAFAgh4VfASUAQIQMB6AmPGjBEDA+kTEhMTaceOHVSrVi1pMvYhoDqBtm3b0s8//ywbbMmF5O4NZcuWFfv76ibgFb6IJty/T8l/PqPU6Ghy9PcnFyGwd/LHgjCq+8PUYIEQ8GrwpaHIEIBA1gUqVKhAMTExshuKFi0qTvMkbfWVXYADCKhEgFcH/Oijj+jChQuKEnHQ+8EHH9CePXtk3XUUF6o4IVXoWhR7+hQlHD9MKVeFOsZFyUvr4EgOZauQW/c+5NG8BTk4O8vP4wgCWRRAwJtFKFwGAQhoT2D9+vXEi01IA1sOEnr37k2ff/659iqEEtulAPc379mzJ127dk0xewP/WtG4cWNau3atohVY1VhCa250yA6KXx5MFPM6a0V19yaPeUvJo36DrF2PqyAgEUDAK8HALgQgoC+BEiVKKFq+IiMjxSWGzS1Aoa/aozZ6EuB+5zyQ7caNG2arxd1zNm3apPh7N3uxjRPjLn5PMXOmEL18moOSOJDLgE8o34hRObgXt9izAAJee377qDsELCjA/yBz66laVizjAT3/+Mc/ZAN+uHydO3emRYsWWbDmyAoCeSPAXXM46OU+vc4mP+1zK3CTJk1ow4YNeVOYnDxF6J4RsWAeJe3ZJNydaj4HJ1dyLF+VHN8vRymPH1HK/dtCYPxMuDxFdr3rsPHkPWiILA0HEMhIAAFvRjo4BwEIaFagf//+in6PPLiHl2jlll9sENCaAH+p5L/h5s2bi9PsmQ5W4/M8t/RXX30l+6KnhnomCQPRIocPotQ/7pkpjgM51mhEbt16knvtuuTg4iK7JlGYUjBq+kRKvXdTku5AHgtXkkfjppI07EIgfQEEvOnb4AwEIKBRgdevX9P7779PHh4eshoUK1aMTp8+rYmffWUFxwEEJALcmtuoUSN69uyZ4m+ZB7nx3NLfffed5A7b7ib8couihvRWDkjjYvkWJM85XwiBbp2MCykE86/HjaHks6Fp17l6kt+RC+To65uWhj0IpCOAgDcdGCRDAALaFeBR7SdPnpT97BsbG0uLFy+mHj16aLdiKDkE/ifAf8/c0vvkyRNF0Mtdd7p27UrBwcKAMBtvcZcuUszYwURJCcqSOLmQ97oQci1XXnkunZTw/r0o9eerxrNOLbuS7+fzjcfYgUB6Agh405NBOgQgoEkBbvXi1i+enF+68UwNt27dkiZhHwKaFuDAtmrVqsS/aJhu3NLL/x3wTCW22mKPH6PYqZ8IkwYnZ1AEoTtD3eaUb/pscgrIn8F1f59KDvuLXrdtRJTwv6kGHZ3J78QPaOXNVA4XIODF3wAEIKArAR6pPmvWLFmduG/jtGnTaMCAAbJ0HEBA6wJhYWHUtGlTevXqlaLfLgfEPJ3ZvHnz8rya8f/vJkUPFn5NSZF/8SRnV2EAmjBgLTlRXiaPfOT5xUpyr5H5Ut+Rq1dR4uovjPe7T19Enu07GI+xAwFzAgh4zakgDQIQ0KxAgwYNxJ95pRVwc3MTl2jl5YSxQUBvAtynt0aNGhQREaGoWnx8PAUFBdE///lPxTlrJfAAtYjA1so+uz75Kd/6neRcsCBFbVhHiZtXESXGpRXD0Yk8gr8hjwYN09LM7KXGxVF4w8p/d5MQFqbwWrOT3D740MyVSIJAmgAC3jQL7EEAAhoX4H67/I+7dI5dbuWqXLky7d69W+O1Q/EhkL7An3/+SXXr1jV7Af/CMWTIEJo4caLZ85ZODO/USpiN4b482wJvku/2vbJlgpP/+4IiRg2l1Ac/pV0rdFEQ+/WWr5CWZmYvYs5MSjqym9xnBJNni5ZmrkASBOQCCHjlHjiCAAQ0LMArTj169Ej2025UVBRdvHiRSpUqpeGaoegQyFzgxYsX1KpVK7F7g+nV0dHRYsDLc1Nbc4sJPUhxM8fIH+FTgHxDQskpfwF5Oh8JfY1fffwRpfz4fdo5vzfIP/Q0OQj97rFBwFICCHgtJYl8IAABmwrcu3ePOnToQPzzrnTj1t2QkBBpEvYhoFsB/sLHg9UcHR0VdeRliOfMmUP9+vVTnLNIgvBrSljT2kQRL9OyE2ZiyLf9ELmULJmWZrKXKgwwDW/VgCj8ufGMc4/B5PPZBOMxdiCQWwEEvLkVxP0QgIAqBLiP4urVqxX/0POqah07dlRFGVEICOSFwG+//SauKGiuT2+c0P91ypQpNHz4cIsXJeHuXYrq3UaWr+uoKeTdL/PBorHnzlLsp4PS7nX1IP+z/yIHV2GQGzYIWEAAAa8FEJEFBCBge4GiRYsqFppwEVZsunbtGuXLl8/2BUQJIJCHAtevX6c+ffqIv3iYrsjGc/iuXLlS/EXEkkWK3h1C8f+clJaltx8FnPqBhPXG09Iy2AtrKbTyvnxqvMLrW2EwWiVhcBo2CFhAAAGvBRCRBQT0IJAijPB2EFYmM13WUwt1W7hwIS1fvpw4wDVs3LVh2LBhYmuWIQ2fELAngYcPH1Lbtm2JZ2ow3XggG0/fx4M8LbVFLJhLSSHrjdk5NWpHvsFfGo8z2wnv2ZlS7//beJnr2Jnk3auP8Rg7EMiNAALe3OjhXgjoRCD21EmKnSj8xOnsQk4N25BHUD9yLVtOE7Xjn2h5HlIepS7duFXrrvATq1MWW5ek92IfAnoRuHTpkthnlxeiMG3p5UB42bJlYvcHS9Q3atsWSlg805iVc/dB5DMuazNDpApfUMMbVSWK/9+CEkIuLv1HUb6Ro435YQcCuRFAwJsbPdwLAT0I8ECTJjWJIsPktSlcnFy79iLPLt3IUcVdAn755Rdq3bq1IrBt3749ffll1luX5JXHEQT0I3D79m2xH7vp6oNcQw6Cly5dSm3ayPve5qT2ij68BYpSwJFz/JBMs4ta+y0lrFwgu8597nJMOSYTwUFuBBDw5kYP90JABwIxRw5T3DRh+c/0NmFeTMcajci9d19yr1krvatslt6lSxf68ccfZYPVuDVr8+bNVK9ePZuVCw+GgJoELl++TIGBgcSLsJhuPFc1dwvi87nahG4SYfWFPrexkcZs3Cb9k7y6ZJxv7GnhF6YJI4QV2CRLEAsrrwWcvZ7l/r/GB2IHAukIIOBNBwbJELAXgfABvSn1J2FgSVY234Lk0rE7efbsTU4FCmZ6R3J4GMXu3UOOhQqTu7B6kqO3d6b3ZOcCnoqMg1pvSb7cN9HPz08MgrOTF66FgN4FDN0bOMA1t61Zs0ac0szcuaymxezfR3FzPku73NWTvJauIbdq1dPSJHvRITsoPni6sASxJNgVzjt3+4h8xksGwEnuwS4EciKAgDcnargHAnoR4BaZmmWV69pnVj9hOU+H8tXIvVc/8mjSlITmVbN3vBo+mFKunvn7nLs3eX21zqKjrkeOHEmHDx+W9U3kEei7du2i2rWF+UCxQQACMoErV66IszNIvyQaLuBAmLs3cHeg3GzhvbpS6r2baVkI/3/hMngsefcfKE4zlir8Nxp37SrFLV9MqQ9vpV1n2BNWZfPbeZAcfXwMKfiEQK4FEPDmmhAZQEC7AqnCRPThtd7PXQWEnx6dW3UmD6HLg0vx4sa8Ul6/plfNhFYdScuNc88h5PPpeOM1udnhATd16tSh8PBwWTZFihSh8+fPy7o4yC7AAQTsXIC/JPIMJtJZTQwkPMiTuwPVqFHDkJTtz6RnTymihxA0x7yW3ysEvuQpTBEYI3R5SDXfykw++cl3p7AqWxZ+QZJnjiMIZCyAgDdjH5yFgO4FwlrWF+a+fCavZ8G3yLFMeUq5dCobrb8O5FCyLLl270OebdtT9OaNlLBqoSxfSw5C4VbcMWPGkKtkYnruzsDTLPF0S9ggAIH0BY4dO0YDBw4kdzPL98bExFBoaChVqVIl/QwyOZP4++8UGdRJCG4jMrlSclpo2fVZu5Wc33xLkohdCFhGAAGvZRyRCwQ0K5D44AFF9hZaY5LlS/K6jplBnh07U8ze3ZS4cwulPnuY9Tq6CANjkpNkrbskdGkIOP9jut0fsp7531eWK1eOuPuCdKolXlnq559/pkKFCmU3O1wPAbsT+O6772js2LGKBVsYgr9I7tmzh95/P+e/ACWH/UWRkz6jlH9dyNjW2Y1cB/+DvAd8ZLH/f8j4gThrjwIIeO3xraPOEDARiDkUSnEzxgipqWlnHJ3I69sd5PbBh2IaTzkUu2UDJZ8KJUqITbsui3tOrbuR7+x5Wbw648sOHjxIo0aNkk1FxjMztGjRgr755puMb8ZZCEDAKMDLcfOy3I5m+uFHR0fTmTNnchX08oPihf66/P8dKTeu/N2dgf9/RvgC7PDOu+RU8UPyGjSUnALyG8uEHQhYQwABrzVUkScENCgQMWcmJe3fIi+5ly/57jtBTv4BxnTu9xt7+BDF79gsDEzhVZEkQbLxKnM7DuRYswm59+mXq+nNuNsC/xTL/XSlGw+44bRixYpJk7EPAQhkIsCrFC5YsEDWPchwC7f08n9XBQtmPiuL4Z6MPnnAGi8y4ejrm9FlOAcBiwsg4LU4KTKEgEYFhIAxvIewtOfDn2UVcChVgfy37Tb7U2PS0ycUs3UzJR3aRRRtMkBFlovJQTanN5PeHRUVRRUrVpR1ZeDzpUqVEvsdmhuII70f+xCAgFKAV1zjhVpMW3r5CyZ3Gzp69CiVLl1aeSNSIKARAQS8GnlRKCYE8kKA+9y97thMMdDEuVNf8pkizJWZ3ib8oxh77gzFb90o/Gx5Mb2rlOnCqG3H2k3Jb+kK5bl0Uj799FNx2jFnZ2fjFQlCi9Hs2bNpwIABxjTsQAAC2ROYM2cOrV27VtZVyJADt/Rev36dvLy8DEn4hICmBBDwaup1obAQsL5A/I0fKXpoT/mAM3Ig9zlLybNV60wLEN6xJaU+eZDpdcYLnFwo4IfbxsOMdl6+fEnVqlVTtEJxC9QDYfAdNghAIHcCK1asEFdd4+nJpBu39PoI8+Jy/3l0G5LKYF8rAgh4tfKmUE4I5KFA1KYNlLDsc/kTnV0p37ZQcilZUp4uOUp8+JAiu7UQUiT9eoXBKZQYn+70Zg5FSpD/wROSXNLfXb9+PX3+ubxc3Hd34sSJNHTo0PRvxBkIQCDLAuPHjyeewcFc9yBfoe/t6dOnxdUMs5whLoSACgQQ8KrgJaAIEFCjwKsxIynlwlF50QIKk/++4+Tg6SlP/9/R69kzKPnAVtk57437yKVESYres4sSQ7ZS6tNfZeddho6jfIOzFqy2bNmS7t+/L7vfUygL9y988803Zek4gAAEci6waNEi+vrrrxV95bmlN3/+/OJ/c/yJDQJaEUDAq5U3hXJCII8FUoWVzMI7tyJ6/lj2ZMcq9cnvm3WyNPFAmBYsrH5lYQ6i6LRzbxSjgMNn0o6FvYR79/6e3uz0YSJvH/Lfe4wcPDxk15g7uHnzJjVr1kz8WdVwnv/xrVy5stin15CGTwhAwDIC3NK7ZcsWxTy9/N8dr2i4f/9+KlCggGUehlwgYGUBBLxWBkb2ENCyQNKTPygiUAh6E+Nk1XAZNJbyDRsuS4s5cpjipn0iS3MdMenvyeRlqTk7aNCgAf3xxx+yFieeJ5SXSeWgFxsEIGB5gcmTJ9OOHTvMZswLvPCvK9y3FxsE1C6AgFftbwjlg4CNBWLPnqbYz7jLgaRfrjC7gueyDeReq7axdOF9u1PqL/8yHpOjM/mdvEqOFvjH8M6dO9S+fXvixSWkW5kyZejQoUPSJOxDAAIWFuCZUbZv365o6eX+88WLFxf7+xYuXNjCT0V2ELCsAAJey3oiNwjoUiByySJK3LJSXjc3T/LZc5ycCxWmpOf/oYi29YWYOMV4DS8y4bfcMquepdefkEeU8+pq2CAAAesK8KDQ48ePK2ZI4ady94bQ0FAMZLPuK0DuuRRAwJtLQNwOAXsRCO/fi1J/viqrrsNbpchvVyhFfrWEkraukp3zXLaR3GvXkaXl9KCkMDMETz0m3dzd3enixYv4R1aKgn0IWFGgZ8+edOnSJZLOgc2PM0xZxv89Yp5eK74AZJ0rAQS8ueLDzRCwH4GUyEh61UFYlCLipazSTs06UfKVc0SRYWnpPvkp4PQPace52Fu6dCkFBweTm5ubMZdEYXnjHj16iPOFGhOxAwEIWF2gV69eYtBrOk8vP/j//u//xHl6pf+tWr1AeAAEsiiAgDeLULgMAhAQZlgQ+tJG9euU7py6BiOXoOGUb/RYw2GOP+Pi4ogHq/GCE9ItNjaWnj59qmj1lV6DfQhAwDoCzZs3Fxd6Mf3VhVt6/fz86PLly7IvqNYpBXKFQPYEEPBmzwtXQ8DuBaL37qb4uRPSdxAGtPke/p6cCr6R/jVZPHPjxg0KDAwUfzKV3tKhQwdavHixNAn7EIBAHgnwLyydOnWiW7duKfr0ctDLg0l59hRsEFCTAAJeNb0NlAUCGhF4PW0SJR8JMVtah3LVyH/jdrPnsps4YMAAOnfunKIld+PGjVS3bt3sZofrIQABCwq0atWK7t69q/jvk2dTKVGihDhlGbo3WBAcWeVKAAFvrvhwMwTsVED4By28W3tK/f2uAsB97nLybNFSkZ7dhMePH1P16tUVg2B4sBq3LGGDAARsK8Bdi7h7Q3rdi8qVK0cHDhywbSHxdAj8TwABL/4UIACBHAmkvHpFrz/qIwS9d9Lud/emgHPCXLxOTmlpOdwbOXKkOMeuo6OjMYeYmBhas2YNtWnTxpiGHQhAwHYC3L0hKCiIrl69qmjp5XP8pZUXruD/jk37/Nqu1HiyPQog4LXHt446Q8CCArGnTlL8np2U+uR3cg36iLy6dst17q9fvxaXEf7rr79keRUsWJCuXLkiS8MBBCBgWwEObBs2bEh//vmnIqjlxSlq1apF27Zts20h8XS7F0DAa/d/AgCAgPoEeBL7ESNGKOb77N+/P02bNk19BUaJIGDnAgkJCcTz9N68eVMhwQFx48aNacOGDYpzSIBAXgkg4M0raTwHAhDIskC1atWIW3elP4HyP6g8gI0XocAGAQioTyA6Opp4IBv3vzedp5f/++3evTt98cUX6is4SmQXAgh47eI1o5IQ0I7A0aNHafDgweTq6mosNI/6rlOnDm3dutWYhh0IQEB9AvHx8dS2bVt6+PChonDcvYEDYl4SHBsE8loAAW9ei+N5EIBAugL8D2Lfvn3FieulF/FPoty6y1MdYYMABNQt8N///pc6duwozt5g2tLLAXHv3r1pwYIF6q4ESqc7AQS8unulqBAEtCsQFRVFlStXJm7RlW48vdGePXsUfXql12AfAhBQjwB3YeBVEl+8eKEoFC9Owf19586dqziHBAhYSwABr7VkkS8EIJBtgVmzZtHatWvJxcXFeG9SUhJNnTpV7OZgTMQOBCCgegFu6W3SpAnxF1lpf3wuOP9qw6soLlq0SPX1QAH1IYCAVx/vEbWAgOYF4uLixC4Lnp6esrpwa+9vv/2m+AdTdhEOIAABVQqEh4eLUwyGhYUp/hvmLkwff/wxjR8/XpVlR6H0JYCAV1/vE7WBgGYFVq1aRQsXLpT9o8itu59++il98sknmq0XCg4BexeIjIwUuypxgGu68RfaHj160Pz5801P4RgCFhVAwGtRTmQGAQjkVIBHdt++fVt2Oy8jfOHCBQoICJCl4wACENCWwB9//EEtW7Yk/iXHdONAmL/UjhkzxvQUjiFgMQEEvBajREYQgEBOBX7//XdxCVJvb29ZFh9++CHt3r1bloYDCEBAmwI8t3aNGjWIB62ZbhwIc9A7YcIE01M4hoBFBBDwWoQRmUAAArkRaNasGT148IAcHR2N2fAk9rt27aK6desa07ADAQhoW+Du3bvUvHlzcWCqaeDLxzxAdeDAgdquJEqvSgEEvKp8LSgUBOxHgLsxcHcG6cb/8BUtWpS+//57aTL2IQABHQjcv3+f2rdvTzx1mekWERFBX375JfXp08f0FI4hkCsBBLy54sPNEIBAbgWCg4OJB6xJN+7Tt3r1anF0tzQd+xCAgD4Ebt68Se3atZOtqGioGQ9kW7lypbgqmyENnxDIrQAC3twK4n4IQCBXAhUqVKCYmBhZHj4+PnT69Gny9/eXpeMAAhDQj8C1a9coKChInJPXtFbcpWn58uXUtWtX01M4hkCOBBDw5ogNN0EAApYQ2LhxI02ePJl4NgbDxq07rVu3phUrVhiS8AkBCOhU4Pjx4zRs2DDZdISGqnKXh5CQEKpZs6YhCZ8QyLEAAt4c0+FGCEAgNwK80lKtWrWIJ6aXbrwq06+//kqmMzZIr8E+BCCgH4HLly9T3759ydw8vZz29ddfi1+C9VNj1MQWAv8fAAD//6sJQ74AACt/SURBVO3dB5QV1f3A8R/LLm3poAh4MBjlKEZBqhIQYRH+CNKkKUQRGwqEEgsBpSgIiIBiIUpHpUpvIkSKMUgVS0TEFjUqKLCwsCzVP79J3jL3zVvY8trMfOccz5u5M2/m3s/Feb+duSXfb2cXYUEAAQSiLLB582bp2rWrnDlzJvPKejtq2bKlTJgwITONFQQQ8L7A4sWLpV+/fpKQkOAo7MmTJ2XevHlyww03OPaRgEB2BfIR8GaXiuMQQCCcAr169ZKVK1dKvnz5Mk+rP3YzZ86UG2+8MTONFQQQ8IfApk2brD+CQ5U2f/78MnXqVKlfv36o3aQhcEEBAt4LEnEAAgiEW+DAgQNStWpVSU5ONk5dvHhx2bZtm+iPGwsCCPhPYMGCBdK7d28pXLhwyMJPmTJFbr755pD7SETgfAIEvOfTYR8CCEREoE+fPrJkyRIjsM3IyJBnn31WunTpEpFrclIEEHCHwIoVK0TvEaFaXOpboPnz50v16tXdURhyGTcCBLxxUxVkBAF/CBw8eFBSUlLk0KFDRoFLlCghO3bsMNLYQAABfwosXLhQHnzwQdG3PsGLvgF68803pXbt2sG72EYgSwEC3ixp2IEAApEQeOedd+SBBx4wnu7qdTTt8ccfj8QlOScCCLhQ4PXXX5cnnnhCEhMTjdxru/+kpCTrLVGVKlWMfWwgkJUAAW9WMqQjgEBEBLSn9b59+4zOavrqUjuwXXXVVRG5JidFAAF3CsyYMUMGDBggRYoUcRTg1KlTsn79eqlcubJjHwkIBAsQ8AaLsI0AAhET2LBhg3Tq1MnorKbDkmkHNm23x4IAAggECzz//PMyduxYKVCgQPAuKVq0qPXHcsWKFR37SEDALkDAa9dgHQEEIiagT3Hbt28vO3fuNK5x7Ngx0eGILrvsMiOdDQQQQCAgMHToUJk+fbqjKZTu13F6t27dKuXKlQsczicCDgECXgcJCQggEAmB1NRUa3zdEydOGKfXp7vLli0z0thAAAEEggWGDx8ur776asgnvaVKlZKNGzeGbPoQfB62/SlAwOvPeqfUCERdYNy4cfLCCy8YHVD0qa92VNPe2CwIIIDAhQQGDRoks2bNcszIpveSQoUKWW16L7744gudhv0+FCDg9WGlU2QEYiFQoUIFx2Dy6enp8v333xtBcCzyxjURQMAdAhrY9u/fXxYtWhSyeYMOb7h9+3ajU6w7SkYuIy1AwBtpYc6PAALWa8iRI0caP1Da7k6f7OoTGxYEEEAgJwI6XNkbb7xh3FP0+9oJ9qKLLrI6spUtWzYnp+RYjwsQ8Hq8gikeArEWOH36tLRp00Y+++wzIys6ePyHH35ojNhgHMAGAgggcB6Bbt26WU0YQk1FXr58eWtf8Bi+5zkduzwuQMDr8QqmeAjEWuDf//631K9f39HRpE6dOlZbPB1EngUBBBDIjYD2AZg3b56jTa8+6a1UqZI13GFycjJNHHKD67HvEPB6rEIpDgLxJtCxY0fZsmWL8eoxIyNDpk2bJk2bNo237JIfBBBwkYC26b3rrrusp7nB4/Rq0HvttddaAbF2aGPxtwABr7/rn9IjEFGBPXv2SJMmTYxOafoDpUMIaccSFgQQQCAcAj169JA1a9aEPNXll18uy5cvl4IFC4bcT6I/BAh4/VHPlBKBmAiMGDFCpk6dalxbO6tNmjRJmjVrZqSzgQACCORWQO8rd999tzUWb3Bgq/0IatWqJXPnzjXeNOX2WnzPnQIEvO6sN3KNgCsE6tatK7/++quR19KlS8vatWtFhw9iQQABBMIpoE2otm3bFrLNbrVq1WTOnDmO/gThvD7nil8BAt74rRtyhoCrBRYuXCi9evUyxt7VJy233XabvPjii64uG5lHAIH4FDh16pS0atVK/vWvfxlNqTS3ev/RoHfJkiXxmXlyFVEBAt6I8nJyBPwroJ1FdGIJ+3L06FHZsWOH6CQULAgggECkBFq2bGkFvQkJCcYltCNbw4YNZfr06UY6G94XIOD1fh1TQgSiLvCPf/xDdIxM7aAWWHQ9JSXFar8bSOMTAQQQiJTADTfcIPv27XM0bwiM3rB06dJIXZrzxqEAAW8cVgpZQsDtAn379pVly5YZxdAnLbNnz7Y6jxg72EAAAQQiIKDDHzZv3lx0LPDg8b416NWOs3/7298icGVOGY8CBLzxWCvkCQEXC2g7uYoVKzpmUNPpPtevXy+Mh+niyiXrCLhMQN8s6SQ3+/fvdwS9OrJDo0aNZMaMGS4rFdnNjQABb27U+A4CCGQp0L9/f3nrrbckKSkp85jjx4/LoEGD5KGHHspMYwUBBBCIhsDBgwet5lSHDh1yXE4D4tatW8v48eMd+0jwlgABr7fqk9IgEFOB1NRUaxrhY8eOGfnQue6/+OILI40NBBBAIFoCaWlp1syOe/fudTzp1c61OobvqFGjopUdrhMDAQLeGKBzSQS8KvDOO++IzngU3F5On+w+8sgjXi025UIAARcIaAc2HaHhxIkTjtxqm96ePXtyn3LIeCeBgNc7dUlJEIi5gHYC+fLLL418aGc17Q199dVXG+lsIIAAAtEW+Pnnn+X2228X/Qxe9Emvjh2uza9YvCdAwOu9OqVECMREYPv27XLrrbdK0aJFM6+v7eMuu+wyWbduXWYaKwgggEAsBb799lvRcXo1wA1+G6Ud2Z566ilrWMVY5pFrh1+AgDf8ppwRAV8KtGvXTnbu3Gn8gGi7OQ12r7nmGl+aUGgEEIhPAW3Lq2+k9B4VvOhsbdq84bHHHgvexbaLBQh4XVx5ZB2BeBHQ14NNmjSR4M5qVapUkVWrVsVLNskHAgggkCnwySefSKdOnUTH6w1+0qsjywwfPpwnvZla7l8h4HV/HVICBGIuMHHiRBk9erToaAz2RdvCde/e3Z7EOgIIIBA3Ar/88ovcdNNNITuyaZOsRx99lOEU46a28pYRAt68+fFtBBA4K3DFFVcY0wgrij4h+fTTT6VkyZIYIYAAAnEr8NFHH1lj8WoH2+Anvdq84YknnpD7778/bvNPxrInQMCbPSeOQgCBLASmTJkiw4YNMyaa0I4fnTt3lmeffTaLb5GMAAIIxI+Aji6jHdn03hW86OyRei/r2LFj8C62XSRAwOuiyiKrCMSbgP446CxFu3fvNrKmY1pq+zj7iA3GAWwggAACcSbw8ccfyy233BLyvqX3tKFDh1oTVMRZtslONgUIeLMJxWEIIOAU0OF9dC56fRVoXxo0aCDTpk1zvB60H8M6AgggEG8Cmzdvli5dujiaaGk+9Unvyy+/LC1atIi3bJOfbAgQ8GYDiUMQQCC0wH333Sdr1641Oqtpmzf9UdDXgywIIICA2wR0THENapOTkx1Z15Fo3njjDWncuLFjHwnxLUDAG9/1Q+4QiFuBH374QWrVqmX8KGiv5mLFiol2AmFBAAEE3Crw7rvvWkOSFShQIOTT3tdff13q1avn1uL5Mt8EvL6sdgqNQN4FtKPajBkzjGYL2qb3hRdekLZt2+b9ApwBAQQQiKHAkiVLpE+fPsYbrEB2Dh8+LKtXr5YaNWoEkviMcwEC3jivILKHQDwK6JNcbaf7008/GdkrXbq0rF+/3njqaxzABgIIIOAigYULF0rfvn2NUWgC2ddxxxcvXixVq1YNJPEZxwIEvHFcOWQNgXgV2LBhgzXsWJEiRTKzqEGwttudMGFCZhorCCCAgNsFZs6caY3QEDxGr5ZL2/QuW7ZMatas6fZiej7/BLyer2IKiED4BapVq2bNQW//AUhPT5eNGzfKlVdeGf4LckYEEEAghgKTJ0+WESNGOEak0Szpk169911yySUxzCGXvpAAAe+FhNiPAAKGwHvvvSd33XWXcePXMSrr1q0rc+bMMY5lAwEEEPCKgHZU01nXEhMTjSLp2y1Nmz9/vujDAJb4FCDgjc96IVcIxK1A//79RTtz2BcNePW13h/+8Ad7MusIIICApwR0xrVXXnklZEe2ggULio7uwJPe+KxyAt74rBdyhUBcCuiTjMqVKztu9nqD1/F4CxcuHJf5JlMIIIBAuAR0dJrBgweHfNKrY/cuWrRIrrjiinBdjvOESYCAN0yQnAYBPwgMGTJEpk6dKjo2ZWA5ceKE/PnPf5ZHH300kMQnAggg4GmBUaNGWR10Q/2RX7JkSest2KWXXuppA7cVjoDXbTVGfhGIkUBaWprUrl1bdKxd+6K9lH/88Ud7EusIIICA5wW0acNzzz1njEUeKLQ+6V2zZo2UK1cukMRnjAUIeGNcAVweAbcIaBvdfv36GbMOadvdHj16yOOPP+6WYpBPBBBAIGwCOnKDDsVYtGhR45za/EuD3RUrVoiOT84SewEC3tjXATlAwBUC7dq1c0wZnJSUZHVWYygyV1QhmUQAgQgIDB06VHSsXvswjYHLlChRwnrSW6ZMmUASnzESIOCNETyXRcBNArt27ZJGjRoZM6jpEwxto6bjT7IggAACfhbQfgw6K5uO1GBf9D6pQe+mTZukUKFC9l2sR1mAgDfK4FwOATcKtG/fXrZv326MvattenVazXr16rmxSOQZAQQQCKtA7969rTdeOhFF8FKxYkVZuXKlo+lD8HFsR06AgDdytpwZAU8IfPfdd3LrrbdaU2jaC3TZZZdZY07a01hHAAEE/CzQqVMn2bx5s2PoRn3SW7x4cdmxY4fx4MDPVtEuOwFvtMW5HgIuE5gyZYoMHz7ccZPWIcp0xjUWBBBAAIFzAt27d7ceBgQ/6dWgt0qVKvL222+fO5i1qAkQ8EaNmgsh4E6Ba665RjIyMozMM3e8wcEGAgggYAi0adPG6uSbkJBgpJ8+fVr07ZhO1KOdflmiJ0DAGz1rroSA6wTefPNNeeyxx4zOFqdOnZIWLVpY02u6rkBkGAEEEIiCgI5Xfuedd1p9H4JHb9Anvddee61jivYoZMvXlyDg9XX1U3gEshbQwLZZs2by7bffGgfpjfyjjz6SUqVKGelsIIAAAgiYAp07d5YPPvjA0aZX76M1a9aUuXPnGjNXmt9mK5wCBLzh1ORcCHhI4JtvvpFbbrnFmGhCi9ewYUNremEPFZWiIIAAAhER0KnX9Y3YV199FXKc3gYNGsj06dMjcm1OagoQ8JoebCGAwP8E+vbtaw07Zu94oa/idFahli1b4oQAAgggkA0BfZrbpUsX2bp1q6PzrwbEN998sxX02u+12Tgth+RQgIA3h2AcjoAfBHSM3cqVK1sDptvLm5iYKLt377YnsY4AAgggcAGB9PR060HB119/7WjeoFO0t23bVsaNG3eBs7A7LwIEvHnR47sIeFRg4MCBMnv2bONpxPHjx2XUqFHStWtXj5aaYiGAAAKRE9B+Ec2bNxcNeoMXHb3hpptukhkzZoRs+hB8PNs5FyDgzbkZ30DA0wL6ii0lJUV+/PFHo5xFixaVnTt3cjM2VNhAAAEEsi9w4MABue222+Q///mP40mvBsQ6ssPIkSOzf0KOzLYAAW+2qTgQAX8IaI9inUo4eE74Vq1ayfjx4/2BQCkRQACBCArceOONsm/fPscVtJ+Ezmz54osv8nDBoZO3BALevPnxbQQ8J3DDDTdYN2L72JHHjh2T1atXW2NHeq7AFAgBBBCIssAvv/wijRs3liNHjhhNxzQb2ryhQ4cOMmbMmCjnytuXI+D1dv1SOgRyJPD+++/LHXfcYcwApB0qrrzySnnnnXdydC4ORgABBBDIWkBnsNRhHjX4tT9g0G9o84Zu3brJsGHDsj4Be3IkQMCbIy4ORsDbAr1795aVK1cahdQ2vTr3u04xzIIAAgggED4BbdNbu3Zta7zz4KBXn/R26tRJRo8eHb4L+vhMBLw+rnyKjoBdQMeKrFatmuhoDPalUqVKVsAb3KbXfgzrCCCAAAK5E9AnvE2bNpXDhw87TqD3ZR0TvX///o59JORMgIA3Z14cjYBnBV544QV57rnnjGku9bXaQw89JAMGDPBsuSkYAgggEGuBvXv3Sp06dUTHOg9etOnDo48+agW+wfvYzr4AAW/2rTgSAc8K6A31uuuuszpL2Aupg6V///33IW/C9uNYRwABBBDIm4COz6vj9GpTBh2twb5o2qBBg+T++++3J7OeAwEC3hxgcSgCXhVYuHCh9QTBXj7trKadJoYMGWJPZh0BBBBAIEICOpOlDgGpb9eCl0OHDom+idNpillyLkDAm3MzvoGA5wR0ZIYtW7YY5SpQoIDVgU2nGGZBAAEEEIiOwKeffipt2rRxPOUNXF2bnulUxCw5EyDgzZkXRyPgOQGd8adWrVpSpEgRo2xXXHGFrFq1yjFGpHEQGwgggAACYRfYtGmT9YYt1JPetLQ0a2IKHcGBJfsCBLzZt+JIBDwp0LlzZ9HZ1fLnz59ZPp1oYtKkSdKiRYvMNFYQQAABBKInoOOia7MybV4WvGjazJkzpUGDBsG72M5CgIA3CxiSEfCDgHZIS0lJcXRWu+iii6wg2A8GlBEBBBCIV4Ht27dLx44dQ2ZPn/6OHTvWmgo+5AEkGgIEvAYHGwj4S2DKlCnyzDPPGIXW3sFPP/00HSMMFTYQQACB2AjoxD89e/YMeXF90jtnzhypW7duyP0knhMg4D1nwRoCvhOoXr26aHsw+1KoUCFZs2aNVKhQwZ7MOgIIIIBAjAR27Ngh7dq1M5qeBbKiM7S9/PLL0qxZs0ASnyEECHhDoJCEgB8EFi9ebE0qYe+spmM91q9fX9544w0/EFBGBBBAwDUCes9++OGHpXDhwo48ax+M6dOnS7169Rz7SPivAAEv/xIQ8KmA3hh1dh/7cuTIEfn888+lVKlS9mTWEUAAAQTiQGD9+vXSvXt30ae6wQvNG4JFzG0CXtODLQR8IfDFF19Iy5YtHZ3V9OnujBkzfGFAIRFAAAE3CixYsEAefPBBKVGihCP7SUlJMnv2bLn++usd+/yeQMDr938BlN+XAgMHDrRuigkJCUb5X3zxRbn11luNNDYQQAABBOJLYN68eTJgwADHk1598qujN6xYsUKuueaa+Mp0jHNDwBvjCuDyCERb4MSJE1KpUiVJTk52XHrPnj1MNOFQIQEBBBCIPwEdZWfQoEEh7+U62o42f9B7Pct/BQh4+ZeAgM8EhgwZYnVuSExMzCx5RkaGdePUDhEsCCCAAALuEHjppZdEpxq2TxwUyLlOD6+zZf7ud78LJPn6k4DX19VP4f0mcPToUWvomp9++skout4Yd+3aZaSxgQACCCAQ/wI6bvrkyZPF/hAjkGtt3qBDmpUpUyaQ5NtPAl7fVj0F96PA5s2bRedfD34a0L59exk9erQfSSgzAggg4HoBnXFN+2AE39u1aUPJkiXl73//u+9H3yHgdf0/cwqAQPYFtEOaDjtmH9Lm5MmTsmjRIqlZs2b2T8SRCCCAAAJxJaCd2HTWtVBBb8GCBWXjxo2i08b7dSHg9WvNU27fCeic7K1atRKdSS2w6LiN2qlhw4YNgSQ+EUAAAQRcKqB9NGbOnBmy83HZsmXl/fffD9n0waXFzVG2CXhzxMXBCLhTQF9r9ejRQ9auXWsUQDurLVu2TGrUqGGks4EAAggg4E6BJ554wgp6dUxe+6IzaZYrV86651988cX2Xb5YJ+D1RTVTSL8LaGe1P/7xj5KWlmZQaO/d1atX+/YvfgODDQQQQMAjAr169ZKVK1cazdcCRStfvrw1ZFmoTm6BY7z4ScDrxVqlTAgECUybNk0GDx4sOhpDYNHmDPrU9/HHHw8k8YkAAggg4BGB8z3prVixojU5hXZo88tCwOuXmqacvhaoXLmyo02XNmf45JNPpHTp0r62ofAIIICAVwW6d+9uNWULbt6g5a1atarMnz/f6NfhVQctFwGvl2uXsiFwVkBvaI899pgR8Gpbrg4dOsiYMWMwQgABBBDwsEDv3r1l+fLlxm+AFlf7dlSoUMEKiO2dmb1KQcDr1ZqlXAj8T+DOO+8UHX/XvuiwNevWrRN9rcWCAAIIIOBdAX3Acffdd1vtdnV4MvuiQa92Wp43b54jILYf54V1Al4v1CJlQCALgcOHD8tVV10lhQsXNo7QV1k69q7fOi0YCGwggAACPhLQSYe2bdvmKLEGvb///e+tp8DBAbHjYBcnEPC6uPLIOgIXEujWrZv1JNce2B4/flx0Vh69+bEggAACCPhDQCcZatOmjXz66aeOhx3aibl69erWgxCvahDwerVmKZfvBX788UepV6+eY9Yd/Qv+s88+870PAAgggIAfBXQCIg167TNuqoMGvXXq1JFZs2Y5fje84ETA64VapAwIhBCYNGmSjBo1ytijbblGjx7N011DhQ0EEEDAPwL6O6APQ/bt2+dot6tBr7bpXbBggedACHg9V6UUCIH/CjRo0ED0Ka99SU5OljVr1liz7djTWUcAAQQQ8I+ABr0tW7aU3bt3O570njp1Sm655RaZPHmyp0AIeD1VnRQGgf8KvPvuu9KlSxcpUqRIJkngddXcuXMz01hBAAEEEPCngLbp1SYMhw4dChn0NmrUSKZPn+4ZHAJez1QlBUHgnED9+vWtp7v2Nlqpqany4YcfSqVKlc4dyBoCCCCAgG8FdCSfFi1aON4GKog+BW7btq2MHz/eEz4EvJ6oRgqBwDmBjz/+2JpUQl9L2ZdatWoJT3ftIqwjgAACCGjQ27RpU9m7d6+jTW96errce++9Mnz4cNdDEfC6vgopAAKmwNChQ2XmzJmOV1SvvfaapKSkmAezhQACCCDge4H9+/dbQa++CQy16BTFgwYNCrXLNWkEvK6pKjKKQPYEKleu7PgrXYci27p1q2inNRYEEEAAAQSCBXTUhtatW1tPeu3N4fS4o0ePysMPPyyDBw8O/pprtgl4XVNVZBSBCwuMHDlSXnnlFSlQoEDmwSdOnJCePXvKgAEDMtNYQQABBBBAIFjg559/lubNm4fsyKYzsunvyAMPPBD8NVdsE/C6oprIJAIXFjhy5Ig0btxY9NWUfdGb1FdffeVo4mA/hnUEEEAAAQRUQNvy6rBk+lQ3eNGRHe677z5XPukl4A2uTbYRcKnABx98IHfeeacjsO3YsaPok18WBBBAAAEEsiPw5ZdfWtMQa6e14OYNOnrDU089JXfddVd2ThU3xxDwxk1VkBEE8ibQoUMH2b59u+PmNHv2bGusxbydnW8jgAACCPhJQN8W6oxswSP+qIGO696rVy/5y1/+4hoSAl7XVBUZRSBrgT179lg3puLFi2cepE0ZLrnkEvnnP/+ZmcYKAggggAAC2RX45JNPpFWrVo6O0Pp9DXqffPJJ0REc3LAQ8LqhlsgjAhcQuOeee2TDhg3G092MjAx588035eabb77At9mNAAIIIIBAaIEffvjB6h+iTRmCF23Tq0NhuiHoJeANrj22EXCZgA4a3rBhQ9FP+3LppZdaQbA9jXUEEEAAAQRyKqCzdDZr1kyKFSvm+Kq+TdQ2vV27dnXsi6cEAt54qg3ygkAuBBYsWCB9+/Y1hiLTG5COmfjII4/k4ox8BQEEEEAAAVNg586d0qlTp5BtevWN4sSJE63mD+a34meLgDd+6oKcIJArgSuvvNJqS2X/8vHjx62JJsqXL29PZh0BBBBAAIFcC2zZskVatmwZ8kmv/u5oMzp94xiPCwFvPNYKeUIgmwLz58+3eskmJSVlfkPbWekYipMmTcpMYwUBBBBAAIFwCLz33nui/Ub0TWLwom16dWr7eOw7QsAbXFtsI+ASAe0he8cdd8i2bduMHOsQMjo8WdmyZY10NhBAAAEEEAiHgDal0yHJ8ufP7zidToK0Zs0aue666xz7YplAwBtLfa6NQB4EUlNT5frrr3cMF3PttdeK3oxC3YjycDm+igACCCCAQKbAqlWrrL4iCQkJmWmBlcTERJk1a5bUrFkzkBTzTwLemFcBGUAgdwL9+vWThQsXit5YAou+Tho+fLjrZsAJ5J9PBBBAAAH3CEyfPl2GDRvmePCiJdA2vUuXLpUaNWrERYEIeOOiGsgEAjkT+PXXX62/nO3Brp5BA97vvvsuZyfjaAQQQAABBHIpMHfuXBk4cGDIbxcsWFBWrFghlStXDrk/mokEvNHU5loIhElAh38ZM2aMMdGEBrs6FmK3bt3CdBVOgwACCCCAwIUFtKOaBr0a4NoX7dimnap1ivtYP+kl4LXXDOsIuESgadOm8tVXXxm5TU5OlnXr1kmZMmWMdDYQQAABBBCItMCrr74qo0ePNh7EBK5ZpEgRWb16tVSoUCGQFPVPAt6ok3NBBPImoKMy6NzmhQsXzjyR/hVdp04dmTNnTmYaKwgggAACCERTYN68eaL9SwoVKmRcVn+jNOh966235KqrrjL2RWuDgDda0lwHgTAJpKSkyNdff210EkhLS7P+etZRG1gQQAABBBCIlcC4ceNk7NixjqBX81O6dGlZsmRJTJ70EvDG6l8E10UgFwIfffSR3H777caA3/qXc9WqVWX58uW5OCNfQQABBBBAILwCkydPtvqU2CdFClxBh8xcv3591INeAt5ADfCJgAsEnn76adFhYOyLBrx6c2ncuLE9mXUEEEAAAQRiJvDMM8/IhAkTRPuXBC/lypWzRm8oVapU8K6IbRPwRoyWEyMQfoEqVaqITh1sX0qWLCnvvvuulChRwp7MOgIIIIAAAjEVGDp0qDXVcL58+Rz50Ha+GzdujFpHawJeRxWQgEB8CrzyyisyYsQIo12UDkXWtWtX0b+kWRBAAAEEEIg3gT59+lizfwYPWab51Ce97733XlRmBiXgjbd/GeQHgRACJ06csCaaSE9PN/ZqunZgC9VOyjiQDQQQQAABBGIkoEGvzroWPA2xNsm75JJL5O2335bixYtHNHcEvBHl5eQIhEdg06ZN8qc//cnorKZnbteunTUBRXiuwlkQQAABBBCIjID+hmkThuAZQjXo1Se9+jsXyYWAN5K6nBuBMAnce++91qQS9nZQ+pfyrFmzpHbt2mG6CqdBAAEEEEAgcgK9e/e2RhQKftKrfVOuvvpqWbZsmSMgDlduCHjDJcl5EIiQwP79+0U7q2nnNPui29u3b7cnsY4AAggggEBcC7Rt21Z27tzpaN5w5swZa1KKlStXhpytLa+FIuDNqyDfRyDCAvp0V0dhsP9FrG15J06cKG3atInw1Tk9AggggAAC4RPQwPa+++6zxuK1v7XUK+i+atWqyeLFi60mfMH785ILAt686PFdBCIscODAAdGZ1Q4fPmxcSYcg27Fjh5HGBgIIIIAAAm4R6NChg2zZssXRhEFHH6pfv741nFlwe9+8lI2ANy96fBeBCAusWLFCevbs6RiyRdP69+8f4atzegQQQAABBCIjcOrUKenYsaPoDKKhlptuukmmTZsWaleu0gh4c8XGlxCIjoB2SNM2vPbXOnqTWLdunVx++eXRyQRXQQABBBBAIAICOrSmjiW/efNmx5Pe48ePS5MmTayg1/4bmNtsEPDmVo7vIRBhgdWrV0v37t2NiSa0J2vNmjWtQbwjfHlOjwACCCCAQMQFtAlDq1at5PPPPzf6qgQu3Lp1axk3blxgM9efBLy5puOLCEROQBvud+7c2TEKg3ZW07a75cuXj9zFOTMCCCCAAAJRFNA3l4GgN/hpru5r2LCh1aY3L1ki4M2LHt9FIEICv/76qzRo0ED0dY99ue6662TRokX2JNYRQAABBBBwtYBOPnHs2DFp0aKFfPPNN45+K/oQqEuXLjJ8+PBcl5OAN9d0fBGByAmMGjXKGnbM3kNV/4cfMmSIdOvWLXIX5swIIIAAAgjEUKBevXqyd+9eRw60SV+zZs3k1VdfdezLTgIBb3aUOAaBKApkZGRYE00kJSUZV9WA98Ybb5SCBQsa6WwggAACCCDgFYEjR47Ihx9+aI3DG1wmDXp1OLMxY8YE77rgNgHvBYk4AIHoCuiEEqNHj3a80oluLrgaAggggAAC8Segozfom86cNm8g4I2/uiRHPhbQv161R+quXbt8rEDREUAAAQQQCC2g7X31vwceeED++te/hj4oRCoBbwgUkhCIlcDXX39tdVYrVKhQrLLAdRFAAAEEEIhbAQ12dSQH7eSmAW/v3r2zlVcC3mwxcRAC0RFo06aNNetMQkKCccHA/+BGIhsIIIAAAgj4TMD+e6gjGT377LNyxx13XFCBgPeCRByAQHQEDh486BiZQa+swW/x4sWtv2bT0tKMWdeikzOuggACCCCAQOwEkpOTRTuz6af+JhYtWlSOHj0q+puow3jec889osN2nm8h4D2fDvsQQAABBBBAAAEEXC9AwOv6KqQACCCAAAIIIIAAAucTIOA9nw77EEAAAQQQQAABBFwvQMDr+iqkAAgggAACCCCAgH8FtCObLjp6Q1YLAW9WMqQjgAACCCCAAAIIeEKAgNcT1UghEEAAAQQQQAABBLISIODNSoZ0BBBAAAEEEEAAAU8IEPB6ohopBAIIIIAAAggggEBWAgS8WcmQjgACCCCAAAIIIOAJAQJeT1QjhUAAAQQQQAABBBDISoCANysZ0hFAAAEEEEAAAQQ8IUDA64lqpBAIIIAAAggggAACWQkQ8GYlQzoCCCCAAAIIIICAJwQIeD1RjRQCAQQQQAABBBBAICsBAt6sZEhHAAEEEEAAAQQQ8IQAAa8nqpFCIIAAAggggAACCGQlQMCblQzpCCCAAAIIIIAAAp4QIOD1RDVSCAQQQAABBBBAAIGsBAh4s5IhHQEEEEDAlQKHn31Gzny+K+95L1hQij4xVBIrXpr3c3EGBBCIqQABb0z5uTgCCCCAQDgFfjt2TA42uO7sKX8Ly2mTuveVYg/3Csu5OAkCCMROgIA3dvZcGQEEEEAgzAJnUlMltUmtsJ21QO9BUvTue8J2Pk6EAAKxESDgjY07V0UAAQQQiJDAgRv/IHIyIyxnT3rwUSl2/4NhORcnQQCB2AkQ8MbOnisjgAACCERA4OSePXLyX5/m+MwZz48QOZJqfC958nwpWP16I40NBBBwnwABr/vqjBwjgAACCIRZ4NTen+Vwy5vONv09c+7MZStI6bc3nttmDQEEXCtAwOvaqiPjCCCAAALhEjg8fJicWvy6cboC/YZI0S5/MtLYQAABdwoQ8Lqz3sg1AggggECYBH5LT5eDKbXPtvs9fu6MBQpLqXXbJN/ZoclYEEDA/QIEvO6vQ0qAAAIIIJAHgSNTJsmJiaONMyS2u1uKD3zSSGMDAQTcK0DA6966I+cIIIAAAnkV+O03OZByg8jh/efOlC9Bii/fKInlLjmXxhoCCLhagIDX1dVH5hFAAAEE8iKQvmqlZDz5Z+MUCbVvlpITJxtpbCCAgLsFCHjdXX/kHgEEEEAgDwIHb28hv/17t3GG5EnzpOD1NYw0NhBAwN0CBLzurj9yjwACCCCQS4HjH+2Uo/e2N76dr8LlUmrpO0YaGwgg4H4BAl731yElQAABBBDIhUBqj3vlzLYNxjcLDhotyW1vN9LYQAAB9wsQ8Lq/DikBAggggEAOBU79+B853LqROdFE4WJnhyLbKvkSE3N4Ng5HAIF4FyDgjfcaIn8IIIAAAmEXODTsSTm9bLZx3sQuPaR4v0eMNDYQQMAbAgS83qhHSoEAAgggkE2BM0ePSqpONHHqxLlvJOSXEqs3Sf5Spc+lsYYAAp4RIOD1TFVSEAQQQACB7AikvfY3Ofnac8ahCQ3+T0qOf8lIYwMBBLwjQMDrnbqkJAgggAACFxI4c0YONK4rcuSgcWTR15dKgaurGmlsIICAdwQIeL1Tl5QEAQQQQOACAunLlkrGsP7GUfkqV5VS85caaWwggIC3BAh4vVWflAYBBBBA4DwCB9v8n/z2w5fGEYWeniBFmt9qpLGBAALeEiDg9VZ9UhoEEEAAgSwEjm/fJkcf7GzuLVpKSq/bIpIvn5nOFgIIeEqAgNdT1UlhEEAAAQSyEkh9oJuc2fEPY3fSvf2k2EM9jTQ2EEDAewIEvN6rU0qEAAIIIBAkcOqH7+Vw2xRzoon8SVJy7RZJKFYs6Gg2EUDAawIEvF6rUcqDAAIIIOAQOPTkX+X0qvlGev5b2kqJkWOMNDYQQMCbAgS83qxXSoUAAggg8D+BM2lpktrk7FBkp20TTUg+KTZvtSRdfjlOCCDgAwECXh9UMkVEAAEE/CyQ9spLcnLq8wZBvqtrSqnX5xppbCCAgHcFCHi9W7eUDAEEEEDg9OmzE03UETl6yLAoPHaSFG7YyEhjAwEEvCtAwOvduqVkCCCAgO8F0leukIzBfUyHUuWk9Jr3zTS2EEDA0wIEvJ6uXgqHAAII+Fvg6JxZcvy5wecQ8iVIocHPSZHbWp1LYw0BBDwvQMDr+SqmgAgggIC/BU4fPCBnDqdZc0sklCkrCcnJ/gah9Aj4UICA14eVTpERQAABBBBAAAE/CRDw+qm2KSsCCCCAAAIIIOBDAQJeH1Y6RUYAAQQQQAABBPwkQMDrp9qmrAgggAACCCCAgA8FCHh9WOkUGQEEEEAAAQQQ8JMAAa+fapuyIoAAAggggAACPhQg4PVhpVNkBBBAAAEEEEDATwIEvH6qbcqKAAIIIIAAAgj4UICA14eVTpERQAABBBBAAAE/CRDw+qm2KSsCCCCAAAIIIOBDAQJeH1Y6RUYAAQQQQAABBPwkQMDrp9qmrAgggAACCCCAgA8FCHh9WOkUGQEEEEAAAQQQ8JMAAa+fapuyIoAAAggggAACPhQg4PVhpVNkBBBAAAEEEEDATwIEvH6qbcqKAAIIIIAAAgj4UICA14eVTpERQAABBBBAAAE/CRDw+qm2KSsCCCCAAAIIIOBDAQJeH1Y6RUYAAQQQQAABBPwk8P9fuRQKQtHCagAAAABJRU5ErkJggg==\" data-image-state=\"image-loaded\" width=\"175\" height=\"105\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 136.5px 8px; transform-origin: 136.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIt has integer sides and a prime perimeter.  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 118.5px 8px; transform-origin: 118.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven an integer \"n\" we want to know \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 264.5px 8px; transform-origin: 264.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ehow many triangles can be formed such that the sides are integers and the perimeters are primes less than or equal to the nearest prime to \"n\"\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128px 8px; transform-origin: 128px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe nearest prime number is defined as:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 30.65px; transform-origin: 391px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 70.5px 8px; transform-origin: 70.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\"n\" if \"n\" itself is prime;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 311.5px 8px; transform-origin: 311.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eeither the previous prime before or the next prime after \"n\", whichever has lesser distance to \"n\"; or\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 260.5px 8px; transform-origin: 260.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethe previous prime before \"n\" if the previous and next primes are equidistant to \"n\".\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 380.5px 8px; transform-origin: 380.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAs an example, lets consider \"n = 9\". The nearest prime is 7 because 9 is equidistant between 7 and 11. The primes less than equal to 7 are [2 3 5 7].  There are no integral triangles that can be formed with perimeter 2. For 3 there is one, namely [1 1 1]. For 5 there is one, [1 2 2]. And for 7 there are 2, [1 3 3] and [2 2 3] . So, the total number of prime perimetered integral triangles that can be formed when n = 9 is 4.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 221.5px 8px; transform-origin: 221.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNOTE: Rotations and reflections are irrelevant and counted only once.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function count = numPerims(given_number)\r\n    y = sumPerims(n);\r\nend","test_suite":"%%\r\nn = 7;\r\nc_correct = 4;\r\nassert(isequal(numPerims(n),c_correct))\r\n%%\r\nn = 10;\r\nc_correct = 8;\r\nassert(isequal(numPerims(n),c_correct))\r\n%%\r\nns = 100:200;\r\nc_correct = 571449;\r\nassert(isequal(sum(arrayfun(@(n) numPerims(n),ns)),c_correct))\r\n%%\r\nn = 1000;\r\nc_correct = 1037542;\r\nassert(isequal(numPerims(n),c_correct))\r\n%%\r\nn = 2^16;\r\nc_correct = 180975423920;\r\nassert(isequal(numPerims(n),c_correct))\r\n%%\r\nn = 2^20;\r\nc_correct = 591520654673872;\r\nassert(isequal(numPerims(n),c_correct))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":11,"test_suite_updated_at":"2021-08-20T14:18:59.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-20T09:06:04.000Z","updated_at":"2026-02-09T10:57:16.000Z","published_at":"2021-08-20T14:18:59.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe triangle below is special.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"105\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"175\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"middle\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt has integer sides and a prime perimeter.  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven an integer \\\"n\\\" we want to know \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ehow many triangles can be formed such that the sides are integers and the perimeters are primes less than or equal to the nearest prime to \\\"n\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe nearest prime number is defined as:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\"n\\\" if \\\"n\\\" itself is prime;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eeither the previous prime before or the next prime after \\\"n\\\", whichever has lesser distance to \\\"n\\\"; or\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ethe previous prime before \\\"n\\\" if the previous and next primes are equidistant to \\\"n\\\".\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAs an example, lets consider \\\"n = 9\\\". The nearest prime is 7 because 9 is equidistant between 7 and 11. The primes less than equal to 7 are [2 3 5 7].  There are no integral triangles that can be formed with perimeter 2. For 3 there is one, namely [1 1 1]. For 5 there is one, [1 2 2]. And for 7 there are 2, [1 3 3] and [2 2 3] . So, the total number of prime perimetered integral triangles that can be formed when n = 9 is 4.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNOTE: Rotations and reflections are irrelevant and counted only once.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.png\",\"relationshipId\":\"rId1\"}]},{\"partUri\":\"/media/image1.png\",\"contentType\":\"image/png\",\"content\":\"data:image/png;base64,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\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52614,"title":"Easy Sequences 12: 50th Prime","description":"Write a function that outputs the th prime after a given number .  For example, the th prime after  is .\r\nNOTE: If  itself is prime start counting from the next prime greater than .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 72px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 36px; transform-origin: 407px 36px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 115.5px 8px; transform-origin: 115.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eWrite a function that outputs the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 18px; height: 18px;\" width=\"18\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 106.5px 8px; transform-origin: 106.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eth prime after a given number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 59px 8px; transform-origin: 59px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  For example, the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 18px; height: 18px;\" width=\"18\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44px 8px; transform-origin: 44px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth prime after \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADMAAAAkCAYAAAAkcgIJAAABtklEQVRYhe2XXbGDMBBGj4c6wAAGqqAKcIADHNQCGioBD1hAQyz0PoQdtimbFLYvnZszk4dCvmT/yKZQqVQq/4UWuBpjBJqCvlHzj+LRvnEBAvA0xpTRdsACPICB6HgA+nXdHB6tyWA4IcOK2Li+vyfPe7YgWEZ5tCaSlf6gTjZdjE0n9o31aotGBcrfhEaXpbVhx5bZ9kvaIgswE0vp07RKZJ/AzZjTqDnaaI82i46AjHl9nnNsVvNzGZUMLF/SZll4d0ZGwP7w9ZwcE++Ge7RFrsR039l3Li2F9qRBN6f2FB2vPSfwWnLXkwb1Tu1pLrzW9qDeaYMehXV0/0qdOap10bBlSN8AtEG5mwHE0rWcOap1I11al8TPlZlwMzY+Y5A0P4/WhUQyPevl1PvUIH3D8GhdSGbG5LmUXylquhF/Q+tCNk7PeV37naHVPUXXvEdrclkXtlIo9yPr1JkK7+U02rsZe7S76MY4JCLpMzN2KeijO52jb8Z7VyKPdhd9WkgUBmJUArHESlFp17lBGdWwNdzcNcSj3aUjduJpHSPRoaP/bXq1zmP9/ckaHm2lUqlUKpUSfw3/IxblkdjJAAAAAElFTkSuQmCC\" style=\"width: 25.5px; height: 18px;\" width=\"25.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9px 8px; transform-origin: 9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 25.5px; height: 18px;\" width=\"25.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.5px 8px; transform-origin: 28.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNOTE: If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 193.5px 8px; transform-origin: 193.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e itself is prime start counting from the next prime greater than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = prime50th(n)\r\n  p = 500 / 499 (n - 1) + 229;\r\nend","test_suite":"%%\r\nn = 1;\r\np_correct = 229;\r\nassert(isequal(prime50th(n),p_correct))\r\n%%\r\nn = 500;\r\np_correct = 829;\r\nassert(isequal(prime50th(n),p_correct))\r\n%%\r\nn = 1:1000;\r\ns_correct = 821400;\r\ns_answer = sum(arrayfun(@(i) prime50th(i),n));\r\nassert(isequal(s_answer,s_correct))\r\n%%\r\nn = 1234;\r\np_correct = 1601;\r\nassert(isequal(prime50th(n),p_correct))\r\n%%\r\nn = 1234567890;\r\np_correct = 1234569059;\r\nassert(isequal(prime50th(n),p_correct))\r\n%%\r\nn = intmax;\r\np_correct = 2147484611;\r\nassert(isequal(prime50th(n),p_correct))\r\n%%\r\nn1 = 8123456789101112;\r\nn2 = floor(flintmax * 0.95);\r\np1 = prime50th(n1);\r\np2 = prime50th(n2);\r\np_correct = 16680296081108630;\r\ns_answer = p1 + p2;\r\nassert(p1\u003en1 \u0026\u0026 p2\u003en2)\r\nassert(isprime(p1) \u0026\u0026 isprime(p2))\r\nassert(isequal(p1+p2,p_correct))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":4,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":17,"test_suite_updated_at":"2021-08-26T18:34:48.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-26T04:22:07.000Z","updated_at":"2025-12-01T11:55:26.000Z","published_at":"2021-08-26T07:45:28.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eWrite a function that outputs the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e50\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eth prime after a given number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  For example, the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e50\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth prime after \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e500\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e829\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNOTE: If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e itself is prime start counting from the next prime greater than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52567,"title":"Easy Sequences 7: Easy as Composite Pi","description":"The prime Pi function is defined as the number of prime numbers from 1 to a certain given limit. In MATLAB, it is easy to create a prime Pi procedure, because there are  built-in functions such as \"primes\" and \"isprime\". To calculate the prime Pi up to 100, we may just proceed as follows:\r\n\u003e\u003e numel(primes(100))\r\n\u003e\u003e ans =\r\n    25\r\n\u003e\u003e nnz(isprime(1:100))\r\n\u003e\u003e ans =\r\n    25\r\nCan we make a function for \"composite Pi\", which is the number of composite numbers from 1 to a given limit, inclusive? Let's find out...\r\nNOTE: The number '1' is considered as neither prime nor composite.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 277.6px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 138.8px; transform-origin: 407px 138.8px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 376.5px 8px; transform-origin: 376.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe prime Pi function is defined as the number of prime numbers from 1 to a certain given limit. In MATLAB, it is easy to create a prime Pi procedure, because there are  built-in functions such as \"primes\" and \"isprime\". To calculate the prime Pi up to 100, we may just proceed as follows:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 122.6px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 61.3px; transform-origin: 404px 61.3px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 84px 8.5px; tab-size: 4; transform-origin: 84px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u0026gt;\u0026gt; numel(primes(100))\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 32px 8.5px; tab-size: 4; transform-origin: 32px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u0026gt;\u0026gt; ans =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 24px 8.5px; tab-size: 4; transform-origin: 24px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    25\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 88px 8.5px; tab-size: 4; transform-origin: 88px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u0026gt;\u0026gt; nnz(isprime(1:100))\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 32px 8.5px; tab-size: 4; transform-origin: 32px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u0026gt;\u0026gt; ans =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 24px 8.5px; tab-size: 4; transform-origin: 24px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    25\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 377px 8px; transform-origin: 377px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCan we make a function for \"composite Pi\", which is the number of composite numbers from 1 to a given limit, inclusive? Let's find out...\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 216px 8px; transform-origin: 216px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNOTE: The number '1' is considered as neither prime nor composite.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function cp = compositePi(lim)\r\n    pc = lim * 3 / 4 - 1;\r\nend","test_suite":"%%\r\nlim = 100;\r\ncp_correct = 74;\r\nassert(isequal(compositePi(lim),cp_correct))\r\n%%\r\nlims = 1000:2000;\r\ncp = arrayfun(@(x) compositePi(x),lims);\r\ns = sum(histc(cp,unique(cp)));\r\nassert(isequal(s,lims(2)))\r\n%%\r\nlim = intmax;\r\ncp_correct = 2042386081;\r\nassert(isequal(compositePi(lim),cp_correct))\r\n%%\r\nlim = 1e10;\r\ncp_correct = 9544947488;\r\nassert(isequal(compositePi(lim),cp_correct))\r\n%%\r\nlim = 12345678910;\r\ncp_correct = 11789236852;\r\nassert(isequal(compositePi(lim),cp_correct))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":10,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":"2021-08-19T12:16:47.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-18T06:37:48.000Z","updated_at":"2026-02-09T10:46:24.000Z","published_at":"2021-08-19T11:52:01.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe prime Pi function is defined as the number of prime numbers from 1 to a certain given limit. In MATLAB, it is easy to create a prime Pi procedure, because there are  built-in functions such as \\\"primes\\\" and \\\"isprime\\\". To calculate the prime Pi up to 100, we may just proceed as follows:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[\u003e\u003e numel(primes(100))\\n\u003e\u003e ans =\\n    25\\n\u003e\u003e nnz(isprime(1:100))\\n\u003e\u003e ans =\\n    25]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCan we make a function for \\\"composite Pi\\\", which is the number of composite numbers from 1 to a given limit, inclusive? Let's find out...\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNOTE: The number '1' is considered as neither prime nor composite.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"term":"group:\"Easy Sequences Volume I\"","current_player_id":null,"fields":[{"name":"page","type":"integer","callback":null,"default":1,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"per_page","type":"integer","callback":null,"default":50,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"sort","type":"string","callback":null,"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"body","type":"text","callback":null,"default":"*:*","directive":null,"facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":false},{"name":"group","type":"string","callback":null,"default":null,"directive":"group","facet":true,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"difficulty_rating_bin","type":"string","callback":null,"default":null,"directive":"difficulty_rating_bin","facet":true,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"id","type":"integer","callback":null,"default":null,"directive":"id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"tag","type":"string","callback":null,"default":null,"directive":"tag","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"product","type":"string","callback":null,"default":null,"directive":"product","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"created_at","type":"timeframe","callback":{},"default":null,"directive":"created_at","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"profile_id","type":"integer","callback":null,"default":null,"directive":"author_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"created_by","type":"string","callback":null,"default":null,"directive":"author","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"player_id","type":"integer","callback":null,"default":null,"directive":"solver_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"player","type":"string","callback":null,"default":null,"directive":"solver","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"solvers_count","type":"integer","callback":null,"default":null,"directive":"solvers_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"comments_count","type":"integer","callback":null,"default":null,"directive":"comments_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"likes_count","type":"integer","callback":null,"default":null,"directive":"likes_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"leader_id","type":"integer","callback":null,"default":null,"directive":"leader_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"leading_solution","type":"integer","callback":null,"default":null,"directive":"leading_solution","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true}],"filters":[{"name":"asset_type","type":"string","callback":null,"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":"\"cody:problem\"","prepend":true},{"name":"profile_id","type":"integer","callback":{},"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":"author_id","static":null,"prepend":true}],"query":{"params":{"per_page":50,"term":"group:\"Easy Sequences Volume I\"","current_player":null,"sort":"map(difficulty_value,0,0,999) asc"},"parser":"MathWorks::Search::Solr::QueryParser","directives":{"term":{"directives":{"group":[["group:\"Easy Sequences Volume I\"","","\"","Easy Sequences Volume I","\""]]}}},"facets":{"#\u003cMathWorks::Search::Field:0x00007f2f69ec7b20\u003e":["Easy Sequences Volume I"],"#\u003cMathWorks::Search::Field:0x00007f2f69ec79e0\u003e":null},"filters":{"#\u003cMathWorks::Search::Field:0x00007f2f69ec53c0\u003e":"\"cody:problem\""},"fields":{"#\u003cMathWorks::Search::Field:0x00007f2f69eca1e0\u003e":1,"#\u003cMathWorks::Search::Field:0x00007f2f69eca140\u003e":50,"#\u003cMathWorks::Search::Field:0x00007f2f69ec8520\u003e":"map(difficulty_value,0,0,999) asc","#\u003cMathWorks::Search::Field:0x00007f2f69ec7f80\u003e":"group:\"Easy Sequences Volume I\""},"user_query":{"#\u003cMathWorks::Search::Field:0x00007f2f69ec7f80\u003e":"group:\"Easy Sequences Volume I\""},"queried_facets":{"#\u003cMathWorks::Search::Field:0x00007f2f69ec7b20\u003e":["Easy Sequences Volume I"]}},"query_backend":{"connection":{"configuration":{"index_url":"http://index-op-v2/solr/","query_url":"http://search-op-v2/solr/","direct_access_index_urls":["http://index-op-v2/solr/"],"direct_access_query_urls":["http://search-op-v2/solr/"],"timeout":10,"vhost":"search","exchange":"search.topic","heartbeat":30,"pre_index_mode":false,"host":"rabbitmq-eks","port":5672,"username":"cody-search","password":"78X075ddcV44","virtual_host":"search","indexer":"amqp","http_logging":"true","core":"cody"},"query_connection":{"uri":"http://search-op-v2/solr/cody/","proxy":null,"connection":{"parallel_manager":null,"headers":{"User-Agent":"Faraday v1.0.1"},"params":{},"options":{"params_encoder":"Faraday::FlatParamsEncoder","proxy":null,"bind":null,"timeout":null,"open_timeout":null,"read_timeout":null,"write_timeout":null,"boundary":null,"oauth":null,"context":null,"on_data":null},"ssl":{"verify":true,"ca_file":null,"ca_path":null,"verify_mode":null,"cert_store":null,"client_cert":null,"client_key":null,"certificate":null,"private_key":null,"verify_depth":null,"version":null,"min_version":null,"max_version":null},"default_parallel_manager":null,"builder":{"adapter":{"name":"Faraday::Adapter::NetHttp","args":[],"block":null},"handlers":[{"name":"Faraday::Response::RaiseError","args":[],"block":null}],"app":{"app":{"ssl_cert_store":{"verify_callback":null,"error":null,"error_string":null,"chain":null,"time":null},"app":{},"connection_options":{},"config_block":null}}},"url_prefix":"http://search-op-v2/solr/cody/","manual_proxy":false,"proxy":null},"update_format":"RSolr::JSON::Generator","update_path":"update","options":{"url":"http://search-op-v2/solr/cody"}}},"query":{"params":{"per_page":50,"term":"group:\"Easy Sequences Volume I\"","current_player":null,"sort":"map(difficulty_value,0,0,999) asc"},"parser":"MathWorks::Search::Solr::QueryParser","directives":{"term":{"directives":{"group":[["group:\"Easy Sequences Volume I\"","","\"","Easy Sequences Volume I","\""]]}}},"facets":{"#\u003cMathWorks::Search::Field:0x00007f2f69ec7b20\u003e":["Easy Sequences Volume I"],"#\u003cMathWorks::Search::Field:0x00007f2f69ec79e0\u003e":null},"filters":{"#\u003cMathWorks::Search::Field:0x00007f2f69ec53c0\u003e":"\"cody:problem\""},"fields":{"#\u003cMathWorks::Search::Field:0x00007f2f69eca1e0\u003e":1,"#\u003cMathWorks::Search::Field:0x00007f2f69eca140\u003e":50,"#\u003cMathWorks::Search::Field:0x00007f2f69ec8520\u003e":"map(difficulty_value,0,0,999) asc","#\u003cMathWorks::Search::Field:0x00007f2f69ec7f80\u003e":"group:\"Easy Sequences Volume I\""},"user_query":{"#\u003cMathWorks::Search::Field:0x00007f2f69ec7f80\u003e":"group:\"Easy Sequences Volume I\""},"queried_facets":{"#\u003cMathWorks::Search::Field:0x00007f2f69ec7b20\u003e":["Easy Sequences Volume I"]}},"options":{"fields":["id","difficulty_rating"]},"join":" "},"results":[{"id":52467,"difficulty_rating":"easy-medium"},{"id":52462,"difficulty_rating":"easy-medium"},{"id":52594,"difficulty_rating":"easy-medium"},{"id":52584,"difficulty_rating":"easy-medium"},{"id":52497,"difficulty_rating":"medium"},{"id":52522,"difficulty_rating":"medium"},{"id":52609,"difficulty_rating":"medium"},{"id":52562,"difficulty_rating":"medium-hard"},{"id":52532,"difficulty_rating":"medium-hard"},{"id":52574,"difficulty_rating":"medium-hard"},{"id":52614,"difficulty_rating":"medium-hard"},{"id":52567,"difficulty_rating":"hard"}]}}