{"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":52085,"title":"Compute a sum involving the totient over divisors of a number","description":"Write a function to compute the following sum:\r\n\r\nwhere  is the totient function. The sum is computed over the divisors of  (including 1 and ). The input to the function will be two limits  and . Compute  for . ","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: 116.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 58.2333px; transform-origin: 407px 58.2333px; 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: 143.008px 7.79167px; transform-origin: 143.008px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the following sum:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 35.4667px; 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 17.7333px; text-align: left; transform-origin: 384px 17.7333px; white-space: pre-wrap; 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,iVBORw0KGgoAAAANSUhEUgAAAPwAAABHCAYAAAAjpirRAAAIqklEQVR4nO2da5XqMBSFtwccYAADKEABDnCAAyygAQl4GAtoGAv3/ih79UxI2qRNk7Tsby3+DNMHJTvnkXMCIIQQQgghhBBCCNEOVwD/Gnr9LvtxhfhuTvCL7pr5dQPwfL/GRH9e9BML8eXc8Sm668LXPKAT9sNz7Z+Fry3EV7NDJzJXeMeC17+i8yxKX1uIr+QAv2u/K3gPO/QW/17wukJ8JRd8iv5R8T72Fa69Na5QTkQM4IupLxXu44bl8wjfwKv2DYi22aEbJK7oDxXuRcm7eRwRFxoxh3JD98yVP/kyjvgU/Atl43neh9z66dyRJt4aeRvRCL6CnDUk0u5oO2Y9oQubSogqxZ3fofuOnwvdS4jD+5q5n8cT3bMWCfiKZFoWUwtijwl9juhc5yVFf0RaDoQFWCXzJrnEfkB3/zfn70/UHw+rYo+/a+N0+Vp0s+/4/MJLckQ3wGJLg89Is6YH9MLYY9xVvyPte2LxValczQGdBzJ1LO3R3bMdn664WV+inEQCvtLb1pJpZ9TJMQC9lZrSC/DEsEVl0s3G1jf0g3zo2DF3/vS+PkueXyj3DCnEHKs/NvT0TR5HKC+RzA2foq9pTS07dF9oDdftgm7AndCLMEXwe4xbVSZQH+iTcPxbKKdyxvBk4FpziqZUzcUV+ZYLOTaHzndHnXqS1RIqvW0hKZJz8MyBVj612++BYdeeYrTeAAuTQhbyibCrTLHb747XKDFp5p6gOS6HDBCrSOXaJ3CAP56v6Spx8NQoDHKZKniGTCGBckDb5NZj4Jg9whMIr+Vau6Hz5YY9EzmghxRjfH6wjlWmpjjj08qXXsbx3U+NoiCXqYIHwhaKS2WudfpF2KsZKqXl5GHPxWuUysu8kM+9tuNxzPBcI/9vkB26h+c7yX7gvTXjK72tVQL7QDsbZswR/BN+wdEi20mV7mnIhQ1NBLSGbmKOQkjJyfhWCaiFIbeZ9z7HI7PX4FiMMTrMeySHoUf8zZSGPgDfT1lnPKD/QHNeSxIqva0RH/2irodhmSN4Jp5Cf7cWm/H7CZ1YrRs+VErLAW8nBNsheXqfe8itv7yPf6L3qnbo8wJ011/wC4v3njpWuATHiYkJzJjVCkJPZrJxsgPft5xhN5WIFXzMbjBjrxIWz9dKW3pZjF9gK3HZHMHTyrpCoAtunyuvw52ELEOltMx3cAK5vM/Pa/jOZ4/l/7nGjX/nBEA321evwQksJVfA87mFSjaJHDuBzA5BbQzhu+gdabGR3QZq6qvk0oor+pJLH7RYrXTUzRE8x5EbW3Od3P3fJzrhuRPs2GBmcZA9nrsOhVxdK3Z3cvWFA7YPw50c+IxiodH0GZMpz5veyWRsUsUXA229xdP1SEpa2xTB5wiVxryXOYLPMXmlltLGEvpctgrTJk2t4H2lrrGCt8bU5xFwEkoxMqkTjhcmDny10S+0kUFeCusmlq6+SxFJjlBp7HusLfjUzrgYrOhca03r7iYJ7TFTBW+X23zPxBralPX8LIK3u8TYQUFXacswll+6EcRHikhyhEpjcWcOwc/JXi9RfMQcle8z8T1X1DZ35YYJsYIbe5a23DslH5DS6xDEJrDsbDPFuq8hS0/o0v2ijhezpRh+avaajJXSTsGKyg3VrNvu3rNdwXGJEby17qEQkV516iSXbVWHbi1v8IJp1n0tWXqbyKlZrri1LH1L4Z/tn3AttX3PYicC3/jn5xyyyjYZ7Esk2gkhtZ8jW2LZjeOntpCuJUvPwV2jYcUyO+uakTmCpxvcEtb4uGOZVtx99vYYX4jnW41wsYVdvnPYa6QU0XCiyOIJ0SX7RXfDrXSSLYEtgKhNS0KZI/gftDNxkZB4rYW1OYeh5Tj32CHRDU0a7q7KO/Te5hgMUbJ4pPbDttC5tRR84K240fwSW3CFpwqeImihAcgS6jP3JaltEdrY2PjBsBdqwwXrQbIr0vWmfxBn6VmZl40pbsaaCFU81eYX9b0NO+GnWhE+15aeKfB3nd0+XwqO4rH5nJgQjxNG6PNaD4Jhwwv9uLMewCvymnj/b9Zx0pLlyw1bY0uXz8aQs90yFdau+3IpN8RN/i23bR7QNyjxPik2WupQ1V+ImJbmA/rn+sBfUbNa8Ib4PNlYC3IyR2zXla+9/DYG3cnWXOIYjqi/p0AKdqluTsKWTTilPvcTGZctKYgWN3acS+3ltyPinuvahAP0E1XtlY4UbHw991lnFeEAbBLKMjZSkgZrhPFajUEZm30lbCpZC3e068qHCC3HTaHEjrLcGXeyZ8pBxT7kXDtvtgjjtVqDcsq201esQ0RnrOM+LWP17VPgjr9LhIpM8M06t1sJtyZ3LAVmjmv1AsypPDuj7UmYG1esDbscl9Mq8yfCc4djd2SYSJi1TN3kf00wMVNj+c3+Pnxre99/O1eU3W9BFIDLb6UTYHv0y2tj1VpCiAxwfbTE8hs3PKTV8DUBrSnjLsSqsMtvTErmerkFKTEdf3IbhVgQ39bTNV9bXeYUojpuB1LtVyv7zAshIlhibzUhRKNI8EJ8ERK8EBtjhy4Df8NnPbQEL8RGYZLNrplL8EJsEP4YgNs5JcELsUFYc5/yg4ZCiJXC9lm3DHdM8Pa9g+d4IUQDnNDvJcaNBUM/k+0KnvsIMARgrzKbZeQRCNEQrjVnv7qv5j1k4XnM9X3c3vxtq/sKCLE6KHZb3z4k1JDgrYXn+6zhl1svRAMwMedacgrVtwmIT/D2533ttlXc+loI0QC+HWsp3tCOND7Bc+Kwa/b81d217e0mxCbhpoVuYo7ufGiDSZ/gub2xXcJjl55aYYVoAP58knW5aZUp1As+3Xqf4Okp2P9lWLCDrLwQ1eGWV0zOcUN/ipc72ri4gg+FAJxMVKgjRCPwd7zs74ed0VnnkCvuCpjncDP6XM+XSy/EipHFFuKLkOCF+CIkeCGEEEIIIYQQQgghhBDz+Q8vk1p0mZitNwAAAABJRU5ErkJggg==\" alt=\"y = sum_(d|n) (-1)^(n/d) phi(d)\" style=\"width: 126px; height: 35.5px;\" width=\"126\" height=\"35.5\"\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: 21.0083px 7.79167px; transform-origin: 21.0083px 7.79167px; 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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\" alt=\"phi(d)\" style=\"width: 32.5px; height: 18.5px;\" width=\"32.5\" 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: 20.6083px 7.79167px; transform-origin: 20.6083px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/656\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003etotient 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: 133.4px 7.79167px; transform-origin: 133.4px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The sum is computed over the divisors 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: 53.3px 7.79167px; transform-origin: 53.3px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (including 1 and \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: 83.225px 7.79167px; transform-origin: 83.225px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e). The input to the function will be two limits \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: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \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);\"\u003eb\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: 34.225px 7.79167px; transform-origin: 34.225px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Compute \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);\"\u003ey\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: 12.05px 7.79167px; transform-origin: 12.05px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"a \u003c= n \u003c= b\" style=\"width: 62.5px; height: 18px;\" width=\"62.5\" 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: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; 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 = sumTotientOverDivisors(a,b)\r\n  for n = a:b\r\n      y(n) = sum((-1)^(n/d)*totient(d));\r\n  end\r\nend","test_suite":"%%\r\na = 1;\r\nb = 10;\r\ny = sumTotientOverDivisors(a,b);\r\nX = prod(cumsum(abs(y)));\r\nX_correct = 207360000;\r\nassert(isequal(X,X_correct))\r\n\r\n%%\r\na = 17;\r\nb = 66;\r\ny = sumTotientOverDivisors(a,b);\r\nx = reshape(y(2:end)-y(1:end-1),7,7);\r\nX = trace(x)*trace(fliplr(x));\r\nX_correct = 82369;\r\nassert(isequal(X,X_correct))\r\n\r\n%%\r\na = 97;\r\nb = 123;\r\ny = sumTotientOverDivisors(a,b);\r\nX = char(-y(isprime(abs(y))));\r\nX_correct = 'aegkmq';\r\nassert(isequal(X,X_correct))\r\n\r\n%%\r\na = 14235;\r\nb = 14237;\r\ny = sumTotientOverDivisors(a,b);\r\nX = factor(str2num(regexprep(num2str(y.*sign(y)),' ','')));\r\nX_correct = [383 37167139];\r\nassert(isequal(X,X_correct))\r\n\r\n%%\r\na = 1563423422342437;\r\nb = 1563423422342457;\r\ny = sumTotientOverDivisors(a,b);\r\nX = sum(arrayfun(@(k) sum(num2str(-y(k))-'0'),1:length(y)));\r\nX_correct = 598;\r\nassert(isequal(X,X_correct))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-06-19T03:22:07.000Z","updated_at":"2025-06-27T11:44:27.000Z","published_at":"2021-06-19T03:25:17.000Z","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\u003eWrite a function to compute the following 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:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y = sum_(d|n) (-1)^(n/d) phi(d)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = \\\\sum_{d|n}(-1)^{n/d}\\\\varphi(d)\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=\\\"phi(d)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varphi(d)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/656\\\"\u003e\u003cw:r\u003e\u003cw:t\u003etotient function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. The sum is computed over the divisors 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 (including 1 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\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 input to the function will be two limits \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=\\\"a\\\"/\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 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\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"b\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Compute \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\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for \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=\\\"a \u0026lt;= n \u0026lt;= b\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea \\\\le n \\\\le b\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":51715,"title":"Iterate the sum of divisors and totient","description":"","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: 339px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 169.5px; transform-origin: 407px 169.5px; vertical-align: baseline; \"\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\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46898\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eCody Problem 46898\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: 160.25px 7.91667px; transform-origin: 160.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e deals with the sum of divisors function, denoted by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"sigma(n)\" style=\"width: 30.5px; height: 19px;\" width=\"30.5\" height=\"19\"\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: 21.7833px 7.91667px; transform-origin: 21.7833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, while \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/656\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eCody Problem 656\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: 46.675px 7.91667px; transform-origin: 46.675px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e deals with the totient function, denoted by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD8AAAAmCAYAAABzhkOMAAACkklEQVRoge1Z25GDMAzcHtIBDdAAFaQCOqCDdEAL1JAS6CEtUENa4D7svQifHzIxDHPjneHHEMmSVg87QEVFRUXF4ejtUwoTgKagvMPQw2y2JBoALwBtYblF0QOYD5LdAlhwUQa0AN44dnM9DAMuhwXA4wQ980l61Ohhon47QdcdwIoL0X8BMP5jfUEwEt2JOkcYphVBB5NHspU0AAa7HuvZE4zxWjRW3+Cs360ud92HIg4fYCj0wjaHRiucm1kRbjML0u2ts9+8rawVn1mgtfpX8aQo3Yj97QIj9nTWHx7Br8C3iKz7QKcyauzdA0yx5DtNO8vRuwENX7Ct0PSom0+zWJffd8iLAPVSzoItdWm8Rt4bOwYq0niFyR2JJ7aUJGbxm2+MX4T8Gdv8pixtLjON1Ljhk3cutRh1n3L+5pvIS/kL/lKWQdEaxICowXxe8beC851rYCt+49Isx3jJON9ARGO0eZxtPGnnUx4qanLTbhvKMZ4p5XM8Eu980HSZX0jaxSjvGigd5rY6VmjNtCUp7+Iu3mvHVh8Tg5AFxd2sjK4cdHqEnUJoqq40zieHLZBBuSF+TqDT1X1e5q5rPCkpo8JWlPLwE+kiJft7aFCSxkyI05+BdLtVFKzabl67Uxc3QMNjUSA7YnRlPfE5UQblbvWmboLozKxTpKQxvSYp2VuBE4yjNLlMCobSQtYTH03le221f2HnVVkHY9RsHzlXj1Y5naDFBH8hA4xxD/uE2MFDjWa4IVOKnCI1eZ1CA8OUM461E3bO9C5SlMwBT4hHokPBe8JQi9uLCcfdsLD7ZFX4GNjiit2KwKRPyT8sCNaiYsipsDlI9elcjIXlbarwEf+ElLzJvcxNbUVFRcXl8AOhCgwX4v7BgwAAAABJRU5ErkJggg==\" alt=\"phi(n)\" style=\"width: 31.5px; height: 19px;\" width=\"31.5\" height=\"19\"\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: 164.125px 7.91667px; transform-origin: 164.125px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The sum of divisors is straightforward: for example, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"sigma(12) = 1+2+3+4+6+12 = 28\" style=\"width: 227.5px; height: 19px;\" width=\"227.5\" height=\"19\"\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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The totient 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: 95.2917px 7.91667px; transform-origin: 95.2917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e counts the numbers less 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: 84.4px 7.91667px; transform-origin: 84.4px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that are relatively prime to \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: 44.725px 7.91667px; transform-origin: 44.725px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. For example \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(12) = 4\" style=\"width: 65px; height: 19px;\" width=\"65\" height=\"19\"\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: 68.5083px 7.91667px; transform-origin: 68.5083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e because the greatest common divisor of 12 and four numbers (1, 5, 7, 11) is 1. \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: 384px 7.91667px; transform-origin: 384px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWhat happens if you repeatedly apply the two functions, starting with the sum of divisors and alternating? For example, start with 7. Then \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=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"sigma(7) = 8, phi(8) = 4, sigma(4) = 7, phi(7) = 6, sigma(6) = 12, phi(12) = 4, \" style=\"width: 377.5px; height: 19px;\" width=\"377.5\" height=\"19\"\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: 13.2167px 7.91667px; transform-origin: 13.2167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e etc.\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: 117.458px 7.91667px; transform-origin: 117.458px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand the pattern 7, 6, 12, 4 will repeat. \u003c/span\u003e\u003c/span\u003e\u003c/div\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: 258.3px 7.91667px; transform-origin: 258.3px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eOscillating behavior is plausible because the sum of divisors is always 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: 120.45px 7.91667px; transform-origin: 120.45px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and the totient is always smaller 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: 360.958px 7.91667px; transform-origin: 360.958px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Furthermore, because the totient has a minimum value and the sum of divisors has a maximum value, with enough iterations the sequence would have to hit a repeating pattern.\u003c/span\u003e\u003c/span\u003e\u003c/div\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: 383.4px 7.91667px; transform-origin: 383.4px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes an initial seed and returns the repeating pattern and the index of the sequence where the pattern begins. With an initial seed of 7, the sequence would be 7, 8, 4, 7, 6, 12, 4, 7, 6, 12,… Therefore, the repeating pattern is [7 6 12 4] and the start index is 3.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [q,n0] = sigPhi(n)\r\n%  n  = initial seed\r\n%  q  = vector of repeating pattern\r\n%  n0 = index where the repeating pattern starts (counting the initial seed as index 1)\r\nend","test_suite":"%%\r\nn = 2;\r\nq_correct = [2 3];\r\nn0_correct = 1;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 3;\r\nq_correct = [2 3];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 7;\r\nq_correct = [4 7 6 12];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 12;\r\nq_correct = [12 28];\r\nn0_correct = 1;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 28;\r\nq_correct = [24 60 16 31 30 72];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 101;\r\nq_correct = [72 195 96 252];\r\nn0_correct = 7;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 127;\r\nq_correct = [96 252 72 195];\r\nn0_correct = 7;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 256;\r\nq_correct = [432 1240 480 1512];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 777;\r\nq_correct = [576 1651 1512 4800 1280 3066 864 2520];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 1111;\r\nq_correct = [432 1240 480 1512];\r\nn0_correct = 7;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 5555;\r\nq_correct = [10368 30855 14080 36792];\r\nn0_correct = 23;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 7777;\r\nq_correct = [3024 9920 3840 12264 3456 10200 2560 6138 1800 6045 2880 9906];\r\nn0_correct = 11;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 11111;\r\nq_correct = [3024 9920 3840 12264 3456 10200 2560 6138 1800 6045 2880 9906];\r\nn0_correct = 11;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 77777;\r\nq_correct = [10368 30855 14080 36792];\r\nn0_correct = 27;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 123456;\r\nq_correct = [184320 638898 196560 833280];\r\nn0_correct = 21;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 666666;\r\nq_correct = [1658880 5946666 1801800 8124480];\r\nn0_correct = 39;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 7777777;\r\nq_correct = [191102976000 715162215924 207622711296 859454668800 178362777600 757256331104 283740364800 1100946774480 233003796480 1053092362140 221908377600 1035248323200 204838502400 888208962000 214695936000 952677206208 237283098624 859638312960 185794560000 792731088600 178886400000 749337039360 150493593600 639777817224 152374763520 626874655824 202491394560 925865740800 167215104000 715161022368 219847799808 880002352320 161864220672 609720615224 247328774784 987821856000];\r\nn0_correct = 161;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))","published":true,"deleted":false,"likes_count":3,"comments_count":4,"created_by":46909,"edited_by":46909,"edited_at":"2022-11-28T04:11:02.000Z","deleted_by":null,"deleted_at":null,"solvers_count":17,"test_suite_updated_at":"2021-05-10T14:27:43.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-09T19:28:03.000Z","updated_at":"2026-01-14T13:15:59.000Z","published_at":"2021-05-09T19:36:55.000Z","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:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46898\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 46898\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e deals with the sum of divisors function, denoted 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\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, while \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/656\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 656\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e deals with the totient function, denoted 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\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varphi(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The sum of divisors is straightforward: for example, \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=\\\"sigma(12) = 1+2+3+4+6+12 = 28\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(12) = 1+2+3+4+6+12 = 28\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The totient 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 counts the numbers less 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\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 that are relatively prime 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\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. For example \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=\\\"phi(12) = 4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varphi(12) = 4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e because the greatest common divisor of 12 and four numbers (1, 5, 7, 11) is 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\u003eWhat happens if you repeatedly apply the two functions, starting with the sum of divisors and alternating? For example, start with 7. Then \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=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(7) = 8, phi(8) = 4, sigma(4) = 7, phi(7) = 6, sigma(6) = 12, phi(12) = 4, \\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(7) = 8, \\\\varphi(8) = 4, \\\\sigma(4) = 7, \\\\varphi(7) = 6, \\\\sigma(6) = 12, \\\\varphi(12) = 4,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e etc.\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\u003eand the pattern 7, 6, 12, 4 will repeat. \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\u003eOscillating behavior is plausible because the sum of divisors is always 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\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, and the totient is always smaller 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\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. Furthermore, because the totient has a minimum value and the sum of divisors has a maximum value, with enough iterations the sequence would have to hit a repeating pattern.\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 that takes an initial seed and returns the repeating pattern and the index of the sequence where the pattern begins. With an initial seed of 7, the sequence would be 7, 8, 4, 7, 6, 12, 4, 7, 6, 12,… Therefore, the repeating pattern is [7 6 12 4] and the start index is 3.\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":52085,"title":"Compute a sum involving the totient over divisors of a number","description":"Write a function to compute the following sum:\r\n\r\nwhere  is the totient function. The sum is computed over the divisors of  (including 1 and ). The input to the function will be two limits  and . Compute  for . ","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: 116.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 58.2333px; transform-origin: 407px 58.2333px; 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: 143.008px 7.79167px; transform-origin: 143.008px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the following sum:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 35.4667px; 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 17.7333px; text-align: left; transform-origin: 384px 17.7333px; white-space: pre-wrap; 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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\" alt=\"y = sum_(d|n) (-1)^(n/d) phi(d)\" style=\"width: 126px; height: 35.5px;\" width=\"126\" height=\"35.5\"\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: 21.0083px 7.79167px; transform-origin: 21.0083px 7.79167px; 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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\" alt=\"phi(d)\" style=\"width: 32.5px; height: 18.5px;\" width=\"32.5\" 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: 20.6083px 7.79167px; transform-origin: 20.6083px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/656\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003etotient 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: 133.4px 7.79167px; transform-origin: 133.4px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The sum is computed over the divisors 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: 53.3px 7.79167px; transform-origin: 53.3px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (including 1 and \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: 83.225px 7.79167px; transform-origin: 83.225px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e). The input to the function will be two limits \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: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \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);\"\u003eb\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: 34.225px 7.79167px; transform-origin: 34.225px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Compute \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);\"\u003ey\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: 12.05px 7.79167px; transform-origin: 12.05px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"a \u003c= n \u003c= b\" style=\"width: 62.5px; height: 18px;\" width=\"62.5\" 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: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; 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 = sumTotientOverDivisors(a,b)\r\n  for n = a:b\r\n      y(n) = sum((-1)^(n/d)*totient(d));\r\n  end\r\nend","test_suite":"%%\r\na = 1;\r\nb = 10;\r\ny = sumTotientOverDivisors(a,b);\r\nX = prod(cumsum(abs(y)));\r\nX_correct = 207360000;\r\nassert(isequal(X,X_correct))\r\n\r\n%%\r\na = 17;\r\nb = 66;\r\ny = sumTotientOverDivisors(a,b);\r\nx = reshape(y(2:end)-y(1:end-1),7,7);\r\nX = trace(x)*trace(fliplr(x));\r\nX_correct = 82369;\r\nassert(isequal(X,X_correct))\r\n\r\n%%\r\na = 97;\r\nb = 123;\r\ny = sumTotientOverDivisors(a,b);\r\nX = char(-y(isprime(abs(y))));\r\nX_correct = 'aegkmq';\r\nassert(isequal(X,X_correct))\r\n\r\n%%\r\na = 14235;\r\nb = 14237;\r\ny = sumTotientOverDivisors(a,b);\r\nX = factor(str2num(regexprep(num2str(y.*sign(y)),' ','')));\r\nX_correct = [383 37167139];\r\nassert(isequal(X,X_correct))\r\n\r\n%%\r\na = 1563423422342437;\r\nb = 1563423422342457;\r\ny = sumTotientOverDivisors(a,b);\r\nX = sum(arrayfun(@(k) sum(num2str(-y(k))-'0'),1:length(y)));\r\nX_correct = 598;\r\nassert(isequal(X,X_correct))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-06-19T03:22:07.000Z","updated_at":"2025-06-27T11:44:27.000Z","published_at":"2021-06-19T03:25:17.000Z","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\u003eWrite a function to compute the following 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:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y = sum_(d|n) (-1)^(n/d) phi(d)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = \\\\sum_{d|n}(-1)^{n/d}\\\\varphi(d)\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=\\\"phi(d)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varphi(d)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/656\\\"\u003e\u003cw:r\u003e\u003cw:t\u003etotient function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. The sum is computed over the divisors 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 (including 1 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\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 input to the function will be two limits \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=\\\"a\\\"/\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 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\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"b\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Compute \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\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for \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=\\\"a \u0026lt;= n \u0026lt;= b\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea \\\\le n \\\\le b\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":51715,"title":"Iterate the sum of divisors and totient","description":"","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: 339px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 169.5px; transform-origin: 407px 169.5px; vertical-align: baseline; \"\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\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46898\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eCody Problem 46898\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: 160.25px 7.91667px; transform-origin: 160.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e deals with the sum of divisors function, denoted by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"sigma(n)\" style=\"width: 30.5px; height: 19px;\" width=\"30.5\" height=\"19\"\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: 21.7833px 7.91667px; transform-origin: 21.7833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, while \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/656\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eCody Problem 656\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: 46.675px 7.91667px; transform-origin: 46.675px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e deals with the totient function, denoted by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(n)\" style=\"width: 31.5px; height: 19px;\" width=\"31.5\" height=\"19\"\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: 164.125px 7.91667px; transform-origin: 164.125px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The sum of divisors is straightforward: for example, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAccAAAAmCAYAAACxk1AhAAAI+klEQVR4nO2d4ZHqOgyFTw/pgAa2ASqgAjqgg+1gW6AGSqAHWqAGWtj7I5xBeBNbtpXE5uqbybw787LBcWwdSVYcwHEcx3Ecx3Ecx3Ecx3Ecx3Ecx3Ecx3Ecx3GcrvkB8LV1IwDsMLZl2LohjuOszgCf+05DnAEcM84/YhQxLXsA38/jhLQI7wHc8JmTJLfvWmAAcMDrGfZ4D8D7ODzgM8bXCXlztzUOGO3P9Xmctm1ONvvnoeUL4z1yHOb8bYvsMI6/knm1w7td0WjDqpwxNkzDEcAdwC90N7EX54fHBfFOPGIUyE8ht+9a4QTggelnqB03W3PE9D08ME7OXvnGay71xjdez+Qb/Tlbe4xi/gudoO/E+eFxQ182ARht9w+m7yc1rwaMuvPA2Hd0MHi9lDaswgk6AZoSOU30x3PpFYYG6op4J5wxdljPlPRdK9D4PvB6huFEaP35HDG284z3SSjHYi/PQ/KFd0ezF3YYbQ7HVW+iOCVyKXEc8Bpvt+ffhzbhjr7GIfvg/vw3n6k8pgRyEOdOPXvO16t9k/Xsno1IhfUHvELlC3QGZcCr08IOCL342MDioOo19VDSdzXQ+Ft4XTS+4frvgJdoWt/PgPw0VYwdxvEz1T4pLpYCv8PY/iUNHefXUuK41D184V0klowOvjDeg6X47jDaK/5XK44XTI9DLh/15uTw3sN0/g7vdu6Ov8+YfxsTP47tzew+8/w5SKMYmzhMIc4NfnoHGg/h53mt3tH2XQ2W1z8jPlnP4ves0qvMNjyMrsdocQ4aJkujtEaq84L3SN76t5a4Bynoa0SMOSnPEmRmLPYbDELm5iT/P/tl83Sigjvi9ywdt3D+acbs0s8uCh9I7nqL1sBfFdemB5kyhKVtbY3exPEX8Ym6RFrPWhxTnicnoeXa6dLiyMyLTO/1II4yoljD6LUijt9IZybWyCpZwWg3RiyqlunYOftCp3UTm8+Fz1y0Bv6EtAfETtIYwhv6STnM0ZM4sgJN+3utimMKOmiW3vqS4sg0MY1GL+IohSRmFC1pRRw1ld1r2AYrNMsesb6RwjnlNDAYMhknYTns1BF2ONcDc7F8iDkTu1TMW6Inccz9PSsDtKY4cpJae6dLiuMNY6qY9CKOMgXPNWxps5ZYW2pFHDXIwrdPQPZNaItkcVJY08BineoqcllKnDqkAlOZS1JJlgaeHaTpBK5Rlkwi+W5bzVG7RvJp4ijTqlaRwFriSGFc4lWUpcTxG3+LWHoRR2mLWJwS2ijr12p6EkemVUsKw+Q7hjWHJZxfc+lXWZjF81inckGlrZWeGEPQB95L7nmcg7/lQy15oFYGnoZVW7Fm1eaao1ZwPk0cOQHC8VXD0uJ4xF+H0vodsyXEkf0StrMHcZQiwmvK+wir160Esidx5P2XiII2QIod1vONYh9bmgkFkmOjytGWi7fS29IKCAd+SRRmZeAp7to2UExLPByryLE2Ovo0caRDZrl+tKQ40sP+wd93su6wq560FkdWeU7N6x7EUVamxyIJ+SwsxlQv4sj+KY3eLCJHyz5iZlKzbHfAX6G+otB+yR0JpgYRRSdVPbWlOMr353JYwgisiUXfpSYCr/+TOK+WmvW6WLs4fh+J8yy2Sws3Z9COx12ibTTKt8R52vnH1zamKBXHNe9BjsvY2JNOv2ZcpZxePttL4rxSp8hCHOX74D28wqGBafNUvzJjQFsVOqtZNlI+jLkX46USzw3cLcWRC64lIqf1RlrFQhxbSKEwHVJqEGrbb+kkyYhFu1VhmCYsPTT9RwMyZzhLxXHNe5DjPnZ+qooxdt2ao3QuWojjGetV764Bo+BUn8pdq8gX3jM6Wf1Crzo2ePaKc7YUxzPKvSQXR33keI6cU5NCoXNTs6OMJnL8TZxnWbjBiEXrNGijrnvivNQY4GsbsciH0ZGM8DRR9Vr3ALynVWNjT9oujdhrI8dr4rxSYaoVR/m+6idApzk1N+WmB2Hfy63laANUSG9nrkNl5Dh34a3EsUYYaypsvVrVBgthTLH2e47A67lY7cLE69VGt6XRnUVUbXUPgM5hD8+z+N2W1xwthbGFalWtMAKvpcG5QCfch1b143wQsehJet5zBpICWrpeVGKAa4QReA3Ekoco27xF+mWqHb2J4xrCCGwrjmfj69Ua+FR0J6MjGeFZRNWW4gi8jF3sepbVn0C74mgdMW691JIjjLK9mq0pVe3SeF9ScWMTvabys8TAM00TE8ZD4noU9C3fc/yfq1WvSAujZlekFFuII9OqVi+iWwtLjB6qVYF3Yzc3RuSao8X4bVEc90gLo3Z3KrJltSo1JyWMUms432JjKyubI/O0c0aKA1CzkMm1jFxyDbzGS6LnEWszf7fnhetexVET9Vt9e9NaHPl1ibm201G0ihoBF8cpZOZrzhDTflnVFbQmjpoIixmaHvaR1maTzsE5XIOO2XyOBXWmSpZXh8gdCTQCckbZGot8lURbkcRS5blDY5xu6P/Dxzl9V4r19TlI+Q26qYML6FavV1iKo9zcPmwfjZX1h1VdHKfh+J8q06d4LpFuXEocZX1H6jc41qY2aZHHA318gUgWzsTuh2n/8Jnyb+f2VmVfqeelbNAPRkNyEI3IGQR8sFojOuDvThaxbX5khZrmiKW0BtgZ3y3I7bsaLMVRrl9r1iwsBMZaHMP1GDpqdLaW8NBdHOPXpQDwC/CcG1fYzoklxfGA96rKG+Zt2NROMLGjtjhmacKK0tQxlQkY8LIvF4z9aTIWwk3GSw3hHbp0UqowYGrwWea7T+jnO2chJX1Xg5U4aopB5GElMkusOYbzxfoDuCFriiPXm3raPD3cdPyIZTIpS4ljbm1C7ppg63Yut4Yj9my5vrr0WMjmgD5EJzcq/p/h52Raf6ZzDNB9EqdluM7ZxCQv5BPu4QvLO0LOB3OBbTGCNd/of63RcRzH6YyWq6OYq3fPz3Ecx1mdHew/21MLRbulNjmO4zj/GQMqPhVizIC/33xzHMdxnM1ooZBjQBvtcBzHcRzHcRzHcRzHcRzHcRzHcRzHcRzHcRzHcRzH+Vz+Ad8nwVEOBsgFAAAAAElFTkSuQmCC\" alt=\"sigma(12) = 1+2+3+4+6+12 = 28\" style=\"width: 227.5px; height: 19px;\" width=\"227.5\" height=\"19\"\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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The totient 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: 95.2917px 7.91667px; transform-origin: 95.2917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e counts the numbers less 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: 84.4px 7.91667px; transform-origin: 84.4px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that are relatively prime to \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: 44.725px 7.91667px; transform-origin: 44.725px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. For example \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(12) = 4\" style=\"width: 65px; height: 19px;\" width=\"65\" height=\"19\"\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: 68.5083px 7.91667px; transform-origin: 68.5083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e because the greatest common divisor of 12 and four numbers (1, 5, 7, 11) is 1. \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: 384px 7.91667px; transform-origin: 384px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWhat happens if you repeatedly apply the two functions, starting with the sum of divisors and alternating? For example, start with 7. Then \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=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"sigma(7) = 8, phi(8) = 4, sigma(4) = 7, phi(7) = 6, sigma(6) = 12, phi(12) = 4, \" style=\"width: 377.5px; height: 19px;\" width=\"377.5\" height=\"19\"\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: 13.2167px 7.91667px; transform-origin: 13.2167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e etc.\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: 117.458px 7.91667px; transform-origin: 117.458px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand the pattern 7, 6, 12, 4 will repeat. \u003c/span\u003e\u003c/span\u003e\u003c/div\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: 258.3px 7.91667px; transform-origin: 258.3px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eOscillating behavior is plausible because the sum of divisors is always 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: 120.45px 7.91667px; transform-origin: 120.45px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and the totient is always smaller 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: 360.958px 7.91667px; transform-origin: 360.958px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Furthermore, because the totient has a minimum value and the sum of divisors has a maximum value, with enough iterations the sequence would have to hit a repeating pattern.\u003c/span\u003e\u003c/span\u003e\u003c/div\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: 383.4px 7.91667px; transform-origin: 383.4px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes an initial seed and returns the repeating pattern and the index of the sequence where the pattern begins. With an initial seed of 7, the sequence would be 7, 8, 4, 7, 6, 12, 4, 7, 6, 12,… Therefore, the repeating pattern is [7 6 12 4] and the start index is 3.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [q,n0] = sigPhi(n)\r\n%  n  = initial seed\r\n%  q  = vector of repeating pattern\r\n%  n0 = index where the repeating pattern starts (counting the initial seed as index 1)\r\nend","test_suite":"%%\r\nn = 2;\r\nq_correct = [2 3];\r\nn0_correct = 1;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 3;\r\nq_correct = [2 3];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 7;\r\nq_correct = [4 7 6 12];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 12;\r\nq_correct = [12 28];\r\nn0_correct = 1;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 28;\r\nq_correct = [24 60 16 31 30 72];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 101;\r\nq_correct = [72 195 96 252];\r\nn0_correct = 7;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 127;\r\nq_correct = [96 252 72 195];\r\nn0_correct = 7;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 256;\r\nq_correct = [432 1240 480 1512];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 777;\r\nq_correct = [576 1651 1512 4800 1280 3066 864 2520];\r\nn0_correct = 3;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 1111;\r\nq_correct = [432 1240 480 1512];\r\nn0_correct = 7;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 5555;\r\nq_correct = [10368 30855 14080 36792];\r\nn0_correct = 23;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 7777;\r\nq_correct = [3024 9920 3840 12264 3456 10200 2560 6138 1800 6045 2880 9906];\r\nn0_correct = 11;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 11111;\r\nq_correct = [3024 9920 3840 12264 3456 10200 2560 6138 1800 6045 2880 9906];\r\nn0_correct = 11;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 77777;\r\nq_correct = [10368 30855 14080 36792];\r\nn0_correct = 27;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 123456;\r\nq_correct = [184320 638898 196560 833280];\r\nn0_correct = 21;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 666666;\r\nq_correct = [1658880 5946666 1801800 8124480];\r\nn0_correct = 39;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))\r\n\r\n%%\r\nn = 7777777;\r\nq_correct = [191102976000 715162215924 207622711296 859454668800 178362777600 757256331104 283740364800 1100946774480 233003796480 1053092362140 221908377600 1035248323200 204838502400 888208962000 214695936000 952677206208 237283098624 859638312960 185794560000 792731088600 178886400000 749337039360 150493593600 639777817224 152374763520 626874655824 202491394560 925865740800 167215104000 715161022368 219847799808 880002352320 161864220672 609720615224 247328774784 987821856000];\r\nn0_correct = 161;\r\n[q,n0] = sigPhi(n);\r\nassert(isequal(q,q_correct) \u0026\u0026 isequal(n0,n0_correct))","published":true,"deleted":false,"likes_count":3,"comments_count":4,"created_by":46909,"edited_by":46909,"edited_at":"2022-11-28T04:11:02.000Z","deleted_by":null,"deleted_at":null,"solvers_count":17,"test_suite_updated_at":"2021-05-10T14:27:43.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-09T19:28:03.000Z","updated_at":"2026-01-14T13:15:59.000Z","published_at":"2021-05-09T19:36:55.000Z","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:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46898\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 46898\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e deals with the sum of divisors function, denoted 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\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, while \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/656\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 656\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e deals with the totient function, denoted 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\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varphi(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The sum of divisors is straightforward: for example, \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=\\\"sigma(12) = 1+2+3+4+6+12 = 28\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(12) = 1+2+3+4+6+12 = 28\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The totient 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 counts the numbers less 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\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 that are relatively prime 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\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. For example \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=\\\"phi(12) = 4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varphi(12) = 4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e because the greatest common divisor of 12 and four numbers (1, 5, 7, 11) is 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\u003eWhat happens if you repeatedly apply the two functions, starting with the sum of divisors and alternating? For example, start with 7. Then \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=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(7) = 8, phi(8) = 4, sigma(4) = 7, phi(7) = 6, sigma(6) = 12, phi(12) = 4, \\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(7) = 8, \\\\varphi(8) = 4, \\\\sigma(4) = 7, \\\\varphi(7) = 6, \\\\sigma(6) = 12, \\\\varphi(12) = 4,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e etc.\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\u003eand the pattern 7, 6, 12, 4 will repeat. \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\u003eOscillating behavior is plausible because the sum of divisors is always 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\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, and the totient is always smaller 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\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. Furthermore, because the totient has a minimum value and the sum of divisors has a maximum value, with enough iterations the sequence would have to hit a repeating pattern.\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 that takes an initial seed and returns the repeating pattern and the index of the sequence where the pattern begins. With an initial seed of 7, the sequence would be 7, 8, 4, 7, 6, 12, 4, 7, 6, 12,… Therefore, the repeating pattern is [7 6 12 4] and the start index is 3.\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":"tag:\"totient\"","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":"tag:\"totient\"","current_player":null,"sort":"map(difficulty_value,0,0,999) asc"},"parser":"MathWorks::Search::Solr::QueryParser","directives":{"term":{"directives":{"tag":[["tag:\"totient\"","","\"","totient","\""]]}}},"facets":{"#\u003cMathWorks::Search::Field:0x00007f30368446b8\u003e":null,"#\u003cMathWorks::Search::Field:0x00007f3036844618\u003e":null},"filters":{"#\u003cMathWorks::Search::Field:0x00007f3036843d58\u003e":"\"cody:problem\""},"fields":{"#\u003cMathWorks::Search::Field:0x00007f3036844938\u003e":1,"#\u003cMathWorks::Search::Field:0x00007f3036844898\u003e":50,"#\u003cMathWorks::Search::Field:0x00007f30368447f8\u003e":"map(difficulty_value,0,0,999) asc","#\u003cMathWorks::Search::Field:0x00007f3036844758\u003e":"tag:\"totient\""},"user_query":{"#\u003cMathWorks::Search::Field:0x00007f3036844758\u003e":"tag:\"totient\""},"queried_facets":{}},"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":"tag:\"totient\"","current_player":null,"sort":"map(difficulty_value,0,0,999) asc"},"parser":"MathWorks::Search::Solr::QueryParser","directives":{"term":{"directives":{"tag":[["tag:\"totient\"","","\"","totient","\""]]}}},"facets":{"#\u003cMathWorks::Search::Field:0x00007f30368446b8\u003e":null,"#\u003cMathWorks::Search::Field:0x00007f3036844618\u003e":null},"filters":{"#\u003cMathWorks::Search::Field:0x00007f3036843d58\u003e":"\"cody:problem\""},"fields":{"#\u003cMathWorks::Search::Field:0x00007f3036844938\u003e":1,"#\u003cMathWorks::Search::Field:0x00007f3036844898\u003e":50,"#\u003cMathWorks::Search::Field:0x00007f30368447f8\u003e":"map(difficulty_value,0,0,999) asc","#\u003cMathWorks::Search::Field:0x00007f3036844758\u003e":"tag:\"totient\""},"user_query":{"#\u003cMathWorks::Search::Field:0x00007f3036844758\u003e":"tag:\"totient\""},"queried_facets":{}},"options":{"fields":["id","difficulty_rating"]},"join":" "},"results":[{"id":52085,"difficulty_rating":"medium"},{"id":51715,"difficulty_rating":"medium"}]}}