{"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":54680,"title":"Determine whether a number is practical","description":"A number  is practical if all smaller numbers can be written as a sum of the proper divisors of . The number 24 is practical because its proper divisors are 1, 2, 3, 4, 6, 8, and 12 and for example\r\n5 = 4+1, 7 = 4+3, 9 = 6+3, 10 = 8+2, 11 = 8+3, 13 = 12+1, 14 = 12+2, 15 = 12+3, 16 = 12+4, \r\n17 = 12+4+1, 18 = 12+6, 19 = 12+3+4, 20 = 12+8, 21 = 12+8+1, 22 = 12+8+2, 23 = 12+8+3\r\nHowever, 23 is not practical because its only proper divisor, 1, cannot be repeated in the sum.\r\nWrite a function to determine whether a number is practical.","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: 153.867px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 76.9333px; transform-origin: 407px 76.9333px; 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: 32.2833px 8px; transform-origin: 32.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA 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: 8.94167px 8px; transform-origin: 8.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is \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: 26.0667px 8px; transform-origin: 26.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003epractical\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: 165.3px 8px; transform-origin: 165.3px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e if all smaller numbers can be written as a sum of the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/ProperDivisor.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eproper divisors\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: 9.71667px 8px; transform-origin: 9.71667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of \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: 60.2833px 8px; transform-origin: 60.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The number 24 is practical because its proper divisors are 1, 2, 3, 4, 6, 8, and 12 and for example\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 40.8667px; 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 20.4333px; transform-origin: 404px 20.4333px; 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: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; 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: 350.35px 8px; tab-size: 4; transform-origin: 350.35px 8px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e5 = 4+1, 7 = 4+3, 9 = 6+3, 10 = 8+2, 11 = 8+3, 13 = 12+1, 14 = 12+2, 15 = 12+3, 16 = 12+4, \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: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; 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: 327.25px 8px; tab-size: 4; transform-origin: 327.25px 8px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e17 = 12+4+1, 18 = 12+6, 19 = 12+3+4, 20 = 12+8, 21 = 12+8+1, 22 = 12+8+2, 23 = 12+8+3\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: 289.783px 8px; transform-origin: 289.783px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eHowever, 23 is not practical because its only proper divisor, 1, cannot be repeated in the sum.\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: 184.633px 8px; transform-origin: 184.633px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to determine whether a number is practical.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function tf = ispractical(n)\r\n  tf = sum(factor(n))\u003cn;\r\nend","test_suite":"%%\r\nassert(ispractical(24))\r\n\r\n%%\r\nassert(ispractical(1))\r\n\r\n%%\r\nassert(ispractical(2))\r\n\r\n%%\r\nassert(~ispractical(3))\r\n\r\n%%\r\nassert(~ispractical(14))\r\n\r\n%%\r\nassert(~ispractical(15))\r\n\r\n%%\r\nassert(ispractical(18))\r\n\r\n%%\r\nassert(ispractical(32))\r\n\r\n%%\r\nassert(~ispractical(174))\r\n\r\n%%\r\nassert(ispractical(544))\r\n\r\n%%\r\nassert(~ispractical(3140))\r\n\r\n%%\r\nassert(ispractical(9044))\r\n\r\n%%\r\nassert(~ispractical(17822))\r\n\r\n%%\r\nassert(ispractical(25650))\r\n\r\n%%\r\nassert(~ispractical(33022))\r\n\r\n%%\r\nassert(ispractical(46170))\r\n\r\n%%\r\nassert(~ispractical(49584))\r\n\r\n%%\r\nassert(~ispractical(56702))\r\n\r\n%%\r\nassert(ispractical(59000))\r\n\r\n%%\r\nassert(ispractical(70866))\r\n\r\n%%\r\nassert(ispractical(83840))\r\n\r\n%%\r\nassert(ispractical(262144))\r\n\r\n%%\r\nassert(~ispractical(1048598))\r\n\r\n%% \r\nassert(ispractical(60466176))\r\n\r\n%%\r\nassert(ispractical(279936000))\r\n\r\n%%\r\nassert(ispractical(21047953604832))\r\n\r\n%%\r\nassert(~ispractical(2*randi(1e6)+1))\r\n\r\n%%\r\nassert(ispractical(prod([2 3 5 7].^randi(6,[1 4]))))\r\n\r\n%%\r\nfiletext = fileread('ispractical.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext, 'read'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2022-05-25T03:47:34.000Z","deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-05-25T03:46:48.000Z","updated_at":"2026-02-02T17:02:08.000Z","published_at":"2022-05-25T03:47:34.000Z","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\u003eA 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\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\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 is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003epractical\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e if all smaller numbers can be written as a sum of the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/ProperDivisor.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eproper divisors\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e of \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\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\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. The number 24 is practical because its proper divisors are 1, 2, 3, 4, 6, 8, and 12 and for example\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[5 = 4+1, 7 = 4+3, 9 = 6+3, 10 = 8+2, 11 = 8+3, 13 = 12+1, 14 = 12+2, 15 = 12+3, 16 = 12+4, \\n17 = 12+4+1, 18 = 12+6, 19 = 12+3+4, 20 = 12+8, 21 = 12+8+1, 22 = 12+8+2, 23 = 12+8+3]]\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\u003eHowever, 23 is not practical because its only proper divisor, 1, cannot be repeated in the sum.\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\u003eWrite a function to determine whether a number is practical.\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":59981,"title":"Compute the Ramanujan tau function","description":"The Ramanujan tau function  is defined by the relation\r\n\r\nwhere . The first few values of  are 1, -24, 252, -1472, and 4830. \r\nWrite a function to compute the Ramanujan tau function.","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: 125px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 62.5px; transform-origin: 407px 62.5px; 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-collapse: preserve; 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: 89.8583px 8px; transform-origin: 89.8583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe Ramanujan tau function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"28.5\" height=\"18.5\" alt=\"tau(n)\" style=\"width: 28.5px; height: 18.5px;\"\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: 77.8px 8px; transform-origin: 77.8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is defined by the relation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 35px; 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 17.5px; text-align: left; transform-origin: 384px 17.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAVgAAABGCAYAAACJx4wfAAASKUlEQVR4Xu2dWaguRxHHk3fF7UkJKtEHxS1gXEiIqLiARISIG0q4qMQYkCCixCASgkRxQSQPLqgEcYkGFVEEFRQURWMERSE+JAFF9ClG0XetX/j+l8rcnume6Z75er5TA8W955yZXqqr/11dXVV98UXxBAeCA8GB4MAqHLh4lVKj0OBAcCA4EBy4KAA2hCA4EBwIDqzEgQDYlRgbxQYHggPBgRTAvsrY8qPOWPNCa89vO2tTNGdfHPhfR80NxaajwVizKamBfppV+HujRw0q/oL9/O/GjXnJobxnJerzVX3LfnhT47qjuLPFgV8fuvtk+/eJI13/h/3+rxVsmZJjX/aLK+qIT3fEgbGV9Hrrw2cH/fiN/bymYADslxtdY/SaBOA+wX73zx3xNpraLwfYpX07IWOfsN99oKLZgPiLBt//2X6+0ejHFeXGpzvlwNRW5ZvWpzcO+lUrgHPYBMh/2EjaxpZ1z2lnvLtPDqSUiFoZSwHsqwNc9ykgLVo9BbCPtwr+5ABO9W0tMB+3it9vxBbrSS06HWUEB4wDqbOGNQA27K1nWNxyg/8C483dA/4AdM822nK7ri0dQPu5Mzxe0fV2HAiAbcfLKGmEAzmA5bMPGt02+B57Eprslg+HXO81WtMOvGV/oq7jciAA9rj8PxO1lwAsjMBtC4H0z832w0c35hJ24U8ahcvWxow/weoCYE9wUCu7hFmUw/2rjDj7+e/h59zBp5TQCw7iSwGWiv9iNHTd2to/FU8DaMsTWcwklxoB7i0eBmPrhalFu0+tjD0BLDLzeaMtzXKnMN6M8UOFChkY90ujnxvhYfJ8I5RIMG/Kg8q7tS4GWJidEkhcUK484YEHXG83wm2slXDL1NGyzFOYDFv3YS8Ay8L+M6M4e5gvIYDmN4y+W8A/FrGnGr3LVePPoMaUSXb3VxyAuApgqVcn+r6rpxoEAHO/stICgosQvpGnvDjNnw7bfrEHgAVcCfqJHc9y2ZBm+pkMyAKUNxjdP6jqXvv5GUZvNhruYlGW3m5E8ArvVAMsdatC345U5ctZcvwvZRJ5vTVlLXMEi9VTjCJC7Tjj3TvAIh9EOpYe6iKzNxm9zSjcGR8pU3I5fZ39eu75jXybhxqsL5NdLgEmTQA2FUqLMfiyBPofZ+rU1wpTsTmvCX4aIDwjWtl3fc+ZoM87rK7YldgCYcAneATTzjPr2bTrEnoGWLXt6QVzSsD6bnsXeyFz8dG7HpnpxoM/bONZfNAcmT8PGAnkCOn323yVhvxfu0Du/34oYOiaik2c1AEcgAmEmwAs9QE8dw74cCqTdo5w18oxpgKc29FkW9l4fZs08OwwCEH+khFbJbYzJZO3tn89f98zwJYu8MxDxvVfRtcdmH3qACuZInkPPvkALDzAuwif/amAJMDyqwdQLJFNwPw+IxYvbwNHdm410jlKc4ClcalQ2rHVo6QzvbxTKtwt2itTBH7Ga9jZ/mPlQiQwIewYc4dWZFxRhvamFn3aSxm9AuycBR4A0Bhqkp8FgNXhE6f7PG89/AsYTp34s6sDLEs1fDDusUZvMZICpJ3nucN8oupVAFbGY7Qh/2wdSttyQmvgtrQpM4iAXWu7mT8Blc+yVuRjBIq0HKcWZfUKsBy2sPWda8I5SwArv1MWE52TKLfElH/+mEaakifkg93e1W4R4z1A+jFG3gyxCsBSGZP4p0beP5ZOr7XlbTGxpsrArsJWa8vMXRKM1j7F8vjwrnQlQrg2j3spv0eARWl50GjJTvAsAaz66j2YtKPOzSN2cH80mopEBYh/cVB8/C5P50/vtL/hX6tHZjcUM35//mC8NNBgalKkQmnXTm241iTFQ4JnrvZQ0x5pmq0j4ySE3n5UKoQ1/dnLtz0CrM42luygzhLAYn9FkXu5kbwCxg6jhvIov9UxMwGL3A+NZFLz36dkJiXv5xW0FgBLBalQ2trMRLUT1butjCVYpg5/2MPA5RYHyiVvLZEeeE6grfPInYZJcosRppOhEIz1iXpL/YlZRbHZYlbAtsp2EveTVxg9zuh3RgCpDgH86acXwlce3qvl81bfazxfe6iQbRoHDWgMjAX/zrFj9wiw2nUsMbOdAsAyJu8zes5Bthljtujajt91kHHwxpu5tPVnDnHYxTPmjiUepw55Zfa8JTE3+I76U+WuZiLQ5BpLbZhT19eanCnTRaouv32WJplbGHjvDUZMdNmfpX3Knvp9+5tOdXOATbtKgJ33ZMJAuOQUPeQ9bQFkh0Ko/iGEnDpj2ilxDytdtXNjWaOhy9vib1aJT16NYOsWgbl25R4BVorKEsVnzwArrRFfUubfx4w4VGKMfGJ08IS5R1Y9L0/aRfM7lJ6pfCVTixg7WEAdTwP/oERBY9GXqwMsjUkJ7DFSG8pOgpr/HSM0GwzhAB62LQZMj49TVvtzAKtvfVQbqyFaJQ9bcoRDd0CVTHwGCE107KBLSSjwYQUgVYfaogHmZ9qCbx799VtN8YV35qR9PDbAavKwGKYOHOiL+D4nnLRHgNU4niWARS5/YISyklqENY+kDJEv4BIj73cv00rJ7REa96HrVcorSvOLf6cUhE0AlkYMQ2mPkauAzn7PyG8X2RpjJpjy/ZwLsGIqGuodRueMtMJpy1I68XMTS5oNdaVWUUXXHYPfXghb/9+fFqcCWaTRU+9cv94A2NajNb88v/tKHezp0I+SS01ouVbMnee58ib/vmSlzFXotypbR3fBPGyRPr2YVrecJjmX8V5DxWbk/Up9IEbJxJ8C2BzIeCEs1b5zY9jD372b2dihjxaWEjPMsE8BsBeOcirXyFxZmGMKyikOfh4tsUun2j53ns/t/yPebw2wWpE4ofMnfFWNrPxYgzjcEoxNuBKQGkayDQFA243S6LYx7wWfJnLMdUeuV/SnlRBWsrzJ536HkNLa/S6hZMwCYPPDsiXA+gVubAHVPGrp+lnjqZHn4OCN1gArjSIHZrMbuvADTcKSAdK7JZPVb01TmrFMEiVl0bWxQ66SLbDqOoa9e+GwZD/zC9iYLHmb2ZLD1B41WPVpybzc2yFXzqzlF9Dc7jMrUO6FGk+NOfU8/O6SgRyrRMJRCiqzG7vgA+9sX+LbWnqaL1BLJbnxglGqUVJvSkNVPWOasNdeW9moSjS9BUMxeUgwLM+DZyrow/N4qd25R4Ctmfx7Alg/fmM7My8DLRU28XiTYKJWAKtGrzXJl0xovtEq6QeRwcV7IJVcJXeaT5m5lVU201KNcuxUMyeEw1smWgqh5/cxvAhyC4v3mlgqcz0C7JgslMj/ngDWKwYp88BwbErOMUp4xDslc7y0rOx7LQBWzCq1N2Yb1egFf/gjgJUj8dDdR1UKHKdWNx+5ltJQZfPVtgZhIQBgzOdU5Q2FyAvZcJVXP2i3fHE3WZEbjU2uGO+aM9x5wMfnun4viXqi/h4BlnaRnOfrRqmUe1N82xPAelvvcPw43MQLiAfPn9a4An9x4VwzFen5caoFWJ30Lt2m5SZazd/9BGIrT/zwLUbkRh0T3pJkEALQlIbqQR2NEod/bkUYA3T6x8TgGSZW9qfo/pRc4Eo/yIkJwC45Ra/h7drfMgmU38LLKOCKwze5evEJLrGtj7W1V4BVsMrc5D97AlivpHiTogKE8G0mdzFPS5OjxrzUdFct5zUAK2bQiF48BjxDPEDp92N+pP67McDTO5r8qa2pn7RosLhvnTM6n/xhMGI5QPe3R9B2gIUTdTScLxrdfShvjmtMtdBsUIA/3GMhIzKOyDlCg0lNRyIOtJuaw49eAXYJCAzNRb3vZoZJ+5lLpAW8wug2I0x4AtglB5hjIrp08Vos8ksB1qcqbMmAxR0Z+VATVdfvKgRvqh4JeMru4yflmM3T+2amEkb4utkqARxjB3AsEh8xol768CujLxshKF4L6HkMloypItdYTHSr5x32fyYik7PFwtIrwMIvFvmSGzXowzuMxCfxGllhES4Nh14yRrXf+JwdLKIsmgpxndolLq1XixA72TFz3dKyR79bCrC9uWO1ZgzAzHUrpfchLalfq/jSe79ybi5L2rSHb0pc10r60TPA1spGSf97fWeN6C36usWcvoCnSwD2mO5YCB40tuVuJTTS0HM3UdbUh5aCHdVHnZWWl/MwKC1nj+/lPAxK+9QzwNIHDo/P4s3DOQ+D0vH17zHW5CDZPE/1XIDVtnSpa8wS5vhv2Dp8yGjuzZBL6lVU2pKbKHP1YRqouVH2VKO3cnyrjd4aTjrkyT+1ByrefUzlzp1jvj21cpLjZ49/bx29teY8zvJvzuB7d6wrreQ1LumbarBW9JKAgWzHC19QGrX3NAR1+vEyoxo3Edmoak7RC1nQ1Ws5F7k5je1dg1VfWsjLHL4c+10dIrfwjFlj/s7iTynAymOAzg+vr51V4cKXASNOzTlhnJNUeWF1j/hMCbZbmSXoS42R3XtHtPYRbMGvtcoY5r2tPSnfC8DCT9p6zxGUmrXGcqxcv4CORXjNadPR+VYCsD6P6DHcsbxTcu2kmjM4vb0LMHNFMUmJ/QPIrmkrPjYfkL+vGSmxttoj9625Dvn6fk8Ae+wxWLt+edMoaEb1odR8yqiVcrN2Py4oPwew3h1rM+dcayX14kiOmxP+jjw1Po+bMzYq7J4DAbDdD9H+G5gDWBnt14pz9xxE4Llb6aVGOBz7m2p5b2lI5P5HKXqwBgf8QaHKr7X7ycPBt3dLxWQNPkWZFRyYAlid5rEVG95RU1Hlw/6l3HujZ7j1S5XNYc7YLZA1bYlvzyYH5LYzXMThhr8TqpQ7/kqf4Tcl15iU1hPv7YwDYwDrjc09dKmFwbuHfkQbjscBhUBz95nMTlOtQZvlGfMgQQM+d3hnaBdPlYuiQqgvz5oBLMfjcNR8AQdSAJuK4T82604tFPTY/DyL9StD19y+j23xa7L/50xzc9sY73fKgRjoTgcmmhUcCA7snwMBsPsfw+hBcCA40CkHAmA7HZhoVnAgOLB/DvQGsDqNJWP9liGx+x/J6EFvHOAs43ajPxgtDYborU/RnpkcaA2wRN3cP7MNvA6wIoTXGhHNcZZCQBewKz5ZmQPII8+SfBvMAWQZ33HcwMIDZuXB6rn41gCr21E5YZ0DtAg091YhnCQyCYDtWWpOv23ykyX3BXlE5wCtLtW8yb7j6pMA2NOXl9EetgZYfAMV3kpo6w0zgVbhiwGwZ1goO+g6Cz4AKS10SfCB3LgCYDsY0GM1oTXAqh9LgTYA9liSEPWmOFADtAGwIVMXlQCsbErcYEoUDBmdHjDCgE8Ey9QKPRdoA2BDKNfkAPLFFT2EayPLVxldbvRpI6K7xvJdLAHaANg1R3InZZcArLqCfZVwPwD2GiMuKOPyOX6Xu2K4FGgDYHciODtupmSMUNg7jC49KAzcYpq7qWMO0AbA7lhIWjW9FGAVPqv4bK5O5rnPqDQDkU99yLepDF0BsK1GNsoZ44DybCjHKKGwyqxVemWMEtDjJUAiolSe5ADYkMEiEwFsklAiTLoFVUJ5s/1u6pYBv+pTFprCXUape7UCYEMo1+aArtvhIPVqI7xdlDkul/MCcxmeBeQqZud2q9FPDmUM2x0Au/ZI7qD8Ug1WeWH9FionlClgxeVlyn0rAHYHQrPzJirpi99BkceVZ+w6pBSwMhem3LcCYHcuKC2aXwqwCOVwKzQmlEuAVX0JgG0xqlHGGAe8fOniTt1Um7oxYwmwqu4A2JDDIhOBhNILoISSVZzDLh62/AgVmgGXI5KkO6exDocgADaEck0OCPS8WUumLuT2ISO2/DyYsjAF6M6znMY6bHcA7JojuZOySzTYlFDKJougXnYAWQAWrRa71FxhFLu42O/Og1BHLoKdCNGOmilTl7e1yiaLixbeMQAtrlvIMe5bS28ARrm4zigCDXYkIK2bWgKw91qllxyAVPZTD4Q32t9qb33UJYeUpZslOdEdOwxrzYco7/Q5gIw9mFi8BYRLIg9TXGN3h+aL8iEvA0Jmxw7DTp/zZ7iHJQA7hz1s8UsfwHpOvoLScuO94EAtBwBjtNjS5x57cU6+gtJy472dc6A1wM65liO2TjsXnhNufupK76nuxo3HJywMNV1rDbA1bYlvgwPBgeDASXEgAPakhjM6ExwIDvTEgQDYnkYj2hIcCA6cFAcCYE9qOKMzwYHgQE8cCIDtaTSiLcGB4MBJceD/ZpTAgx/IMZMAAAAASUVORK5CYII=\" width=\"172\" height=\"35\" alt=\"Sum[tau(n)q^n,{n,1,inf}] = q Product[(1-q^n)^24,{n,1,inf}]\" style=\"width: 172px; height: 35px;\"\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-collapse: preserve; 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: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"88\" height=\"18.5\" alt=\"q = exp(2 pi i z)\" style=\"width: 88px; height: 18.5px;\"\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: 73.8833px 8px; transform-origin: 73.8833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The first few values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"28.5\" height=\"18.5\" alt=\"tau(n)\" style=\"width: 28.5px; height: 18.5px;\"\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: 106.183px 8px; transform-origin: 106.183px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are 1, -24, 252, -1472, and 4830. \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-collapse: preserve; 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: 98.6583px 8px; transform-origin: 98.6583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/TauFunction.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eRamanujan tau function\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: 1.94167px 8px; transform-origin: 1.94167px 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 y = Ramanujantau(n)\r\n  y = q*prod((1-q^n)^24)/sum(q^n);\r\nend","test_suite":"%%\r\nassert(isequal(Ramanujantau(1),1))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(2),-24))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(7),-16744))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(14),401856))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(22),-12830688))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(51),-1740295368))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(86),411016992))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(147),-427635232164))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(243),13400796651732))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(260),4107578522880))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(300),9458784518400))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(325),14731871253050))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(400),-25171202969600))\r\n\r\n%%\r\nassert(isequal(sum(arrayfun(@Ramanujantau,1:300)),-33462718906943))\r\n\r\n%%\r\nfiletext = fileread('Ramanujantau.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext, 'read'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-04-26T13:15:52.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-04-25T03:28:46.000Z","updated_at":"2024-04-26T13:15:52.000Z","published_at":"2024-04-25T03:28:58.000Z","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 Ramanujan tau 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\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"tau(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\tau(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is defined by the relation\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=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Sum[tau(n)q^n,{n,1,inf}] = q Product[(1-q^n)^24,{n,1,inf}]\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sum_{n\\\\ge1} \\\\tau(n) q^n = q\\\\prod_{n\\\\ge1}(1-q^n)^{24}\\n\u003c/w:t\u003e\u003c/w:r\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\u003ewhere \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\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q = exp(2 pi i z)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq = \\\\exp(2\\\\pi i z)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The first few values of \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\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"tau(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\tau(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are 1, -24, 252, -1472, and 4830. \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\u003eWrite a function to compute the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/TauFunction.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eRamanujan tau function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\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\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":54680,"title":"Determine whether a number is practical","description":"A number  is practical if all smaller numbers can be written as a sum of the proper divisors of . The number 24 is practical because its proper divisors are 1, 2, 3, 4, 6, 8, and 12 and for example\r\n5 = 4+1, 7 = 4+3, 9 = 6+3, 10 = 8+2, 11 = 8+3, 13 = 12+1, 14 = 12+2, 15 = 12+3, 16 = 12+4, \r\n17 = 12+4+1, 18 = 12+6, 19 = 12+3+4, 20 = 12+8, 21 = 12+8+1, 22 = 12+8+2, 23 = 12+8+3\r\nHowever, 23 is not practical because its only proper divisor, 1, cannot be repeated in the sum.\r\nWrite a function to determine whether a number is practical.","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: 153.867px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 76.9333px; transform-origin: 407px 76.9333px; 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: 32.2833px 8px; transform-origin: 32.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA 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: 8.94167px 8px; transform-origin: 8.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is \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: 26.0667px 8px; transform-origin: 26.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003epractical\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: 165.3px 8px; transform-origin: 165.3px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e if all smaller numbers can be written as a sum of the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/ProperDivisor.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eproper divisors\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: 9.71667px 8px; transform-origin: 9.71667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of \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: 60.2833px 8px; transform-origin: 60.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The number 24 is practical because its proper divisors are 1, 2, 3, 4, 6, 8, and 12 and for example\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 40.8667px; 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 20.4333px; transform-origin: 404px 20.4333px; 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: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; 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: 350.35px 8px; tab-size: 4; transform-origin: 350.35px 8px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e5 = 4+1, 7 = 4+3, 9 = 6+3, 10 = 8+2, 11 = 8+3, 13 = 12+1, 14 = 12+2, 15 = 12+3, 16 = 12+4, \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: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; 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: 327.25px 8px; tab-size: 4; transform-origin: 327.25px 8px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e17 = 12+4+1, 18 = 12+6, 19 = 12+3+4, 20 = 12+8, 21 = 12+8+1, 22 = 12+8+2, 23 = 12+8+3\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: 289.783px 8px; transform-origin: 289.783px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eHowever, 23 is not practical because its only proper divisor, 1, cannot be repeated in the sum.\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: 184.633px 8px; transform-origin: 184.633px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to determine whether a number is practical.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function tf = ispractical(n)\r\n  tf = sum(factor(n))\u003cn;\r\nend","test_suite":"%%\r\nassert(ispractical(24))\r\n\r\n%%\r\nassert(ispractical(1))\r\n\r\n%%\r\nassert(ispractical(2))\r\n\r\n%%\r\nassert(~ispractical(3))\r\n\r\n%%\r\nassert(~ispractical(14))\r\n\r\n%%\r\nassert(~ispractical(15))\r\n\r\n%%\r\nassert(ispractical(18))\r\n\r\n%%\r\nassert(ispractical(32))\r\n\r\n%%\r\nassert(~ispractical(174))\r\n\r\n%%\r\nassert(ispractical(544))\r\n\r\n%%\r\nassert(~ispractical(3140))\r\n\r\n%%\r\nassert(ispractical(9044))\r\n\r\n%%\r\nassert(~ispractical(17822))\r\n\r\n%%\r\nassert(ispractical(25650))\r\n\r\n%%\r\nassert(~ispractical(33022))\r\n\r\n%%\r\nassert(ispractical(46170))\r\n\r\n%%\r\nassert(~ispractical(49584))\r\n\r\n%%\r\nassert(~ispractical(56702))\r\n\r\n%%\r\nassert(ispractical(59000))\r\n\r\n%%\r\nassert(ispractical(70866))\r\n\r\n%%\r\nassert(ispractical(83840))\r\n\r\n%%\r\nassert(ispractical(262144))\r\n\r\n%%\r\nassert(~ispractical(1048598))\r\n\r\n%% \r\nassert(ispractical(60466176))\r\n\r\n%%\r\nassert(ispractical(279936000))\r\n\r\n%%\r\nassert(ispractical(21047953604832))\r\n\r\n%%\r\nassert(~ispractical(2*randi(1e6)+1))\r\n\r\n%%\r\nassert(ispractical(prod([2 3 5 7].^randi(6,[1 4]))))\r\n\r\n%%\r\nfiletext = fileread('ispractical.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext, 'read'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2022-05-25T03:47:34.000Z","deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-05-25T03:46:48.000Z","updated_at":"2026-02-02T17:02:08.000Z","published_at":"2022-05-25T03:47:34.000Z","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\u003eA 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\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\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 is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003epractical\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e if all smaller numbers can be written as a sum of the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/ProperDivisor.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eproper divisors\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e of \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\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\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. The number 24 is practical because its proper divisors are 1, 2, 3, 4, 6, 8, and 12 and for example\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[5 = 4+1, 7 = 4+3, 9 = 6+3, 10 = 8+2, 11 = 8+3, 13 = 12+1, 14 = 12+2, 15 = 12+3, 16 = 12+4, \\n17 = 12+4+1, 18 = 12+6, 19 = 12+3+4, 20 = 12+8, 21 = 12+8+1, 22 = 12+8+2, 23 = 12+8+3]]\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\u003eHowever, 23 is not practical because its only proper divisor, 1, cannot be repeated in the sum.\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\u003eWrite a function to determine whether a number is practical.\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":59981,"title":"Compute the Ramanujan tau function","description":"The Ramanujan tau function  is defined by the relation\r\n\r\nwhere . The first few values of  are 1, -24, 252, -1472, and 4830. \r\nWrite a function to compute the Ramanujan tau function.","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: 125px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 62.5px; transform-origin: 407px 62.5px; 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-collapse: preserve; 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: 89.8583px 8px; transform-origin: 89.8583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe Ramanujan tau function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"28.5\" height=\"18.5\" alt=\"tau(n)\" style=\"width: 28.5px; height: 18.5px;\"\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: 77.8px 8px; transform-origin: 77.8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is defined by the relation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 35px; 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 17.5px; text-align: left; transform-origin: 384px 17.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"172\" height=\"35\" alt=\"Sum[tau(n)q^n,{n,1,inf}] = q Product[(1-q^n)^24,{n,1,inf}]\" style=\"width: 172px; height: 35px;\"\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-collapse: preserve; 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: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"88\" height=\"18.5\" alt=\"q = exp(2 pi i z)\" style=\"width: 88px; height: 18.5px;\"\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: 73.8833px 8px; transform-origin: 73.8833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The first few values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"28.5\" height=\"18.5\" alt=\"tau(n)\" style=\"width: 28.5px; height: 18.5px;\"\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: 106.183px 8px; transform-origin: 106.183px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are 1, -24, 252, -1472, and 4830. \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-collapse: preserve; 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: 98.6583px 8px; transform-origin: 98.6583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/TauFunction.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eRamanujan tau function\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: 1.94167px 8px; transform-origin: 1.94167px 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 y = Ramanujantau(n)\r\n  y = q*prod((1-q^n)^24)/sum(q^n);\r\nend","test_suite":"%%\r\nassert(isequal(Ramanujantau(1),1))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(2),-24))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(7),-16744))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(14),401856))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(22),-12830688))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(51),-1740295368))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(86),411016992))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(147),-427635232164))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(243),13400796651732))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(260),4107578522880))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(300),9458784518400))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(325),14731871253050))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(400),-25171202969600))\r\n\r\n%%\r\nassert(isequal(sum(arrayfun(@Ramanujantau,1:300)),-33462718906943))\r\n\r\n%%\r\nfiletext = fileread('Ramanujantau.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext, 'read'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-04-26T13:15:52.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-04-25T03:28:46.000Z","updated_at":"2024-04-26T13:15:52.000Z","published_at":"2024-04-25T03:28:58.000Z","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 Ramanujan tau 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\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"tau(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\tau(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is defined by the relation\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=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Sum[tau(n)q^n,{n,1,inf}] = q Product[(1-q^n)^24,{n,1,inf}]\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sum_{n\\\\ge1} \\\\tau(n) q^n = q\\\\prod_{n\\\\ge1}(1-q^n)^{24}\\n\u003c/w:t\u003e\u003c/w:r\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\u003ewhere \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\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q = exp(2 pi i z)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq = \\\\exp(2\\\\pi i z)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The first few values of \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\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"tau(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\tau(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are 1, -24, 252, -1472, and 4830. \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\u003eWrite a function to compute the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/TauFunction.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eRamanujan tau 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