{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-05-26T00:16:20.000Z","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":"2026-05-26T00:00:00.000Z","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,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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,iVBORw0KGgoAAAANSUhEUgAAAEEAAAAlCAYAAAAdkoQJAAACtUlEQVRoge2ZXbHjMAyFPw5lEAIhEASLoAzKIAxKIRgKIRxCoRhK4e6DfdaqN3bsxN3pzuTM9OXGlmT9HMm+cOLEiRPfgQvQN5bZebn/BS7ATBsnXAgH7xvK/ShaOGAAfgGjl9WZbx3wPCj/41iA24H9V+AO/PjfsrKmxzniK0vjjotcC8gJ94yuRyNdzdDhjB4aysrJuwAvXNl8DSbaZcEV54AX+ZSfcGXxFVDkro3kPby8Laf2tMu+w7jhjOm2Fhbi5eWNhWunRnoZvFLbejrcAUfyUZ7Zn5adlz3i6nsg8EFJGzyi+w9uXsjCeyTVpkZCpJ+sR7skdWN0uAi+vK7B27AQ+KAEo1+/u11OXkDcaiTYpuOSWCs+SLWyNQy4Q754j7bNgtL2J1t38YIcEA8dOlQciZl1xpbhJfULLuUlJzbcOqGUZCWvulUqvdc2i5ljspnNnr1O6MmTnlpjDcnWBgEIQ8baSJobUrTnSCbIkWujMITsTH1fwy4nqIbWUk7f4oP2Zk9MgKVG2CinUv3JPn6pnlGkaG0aS5GfLZ/4ciQHbTkhxSmCaruW5KqJ0aZ7rhTig1rHpVrkFpunMknYclIKckLxoGbZN045G23btmwap67I6u979Ma65ST7mJLDg/KZAniv7dgYdQU7fV0IWZAbhjRYpZBzQk8oQ1tWE2V1/mTHlVosH2+UEbY1iq1n8lGRc3O92nYXZZqm1ZisJ8ruA9JbfXGz6S2jLSldcQe2I20JnuQNtymvn57J+ujvpTpHKkvBYiC8As28p6NebOSMUsi5WxkzEi5L8f4bdTdRZVETlNR9CWbqevwRqJSawLbGo17t+PtS9Ak015NqjXvxL16CF9q9YgGhNe4mmBUMbHeUvShtnVWovb+Xoqe9sXp5aoqOwNZf/R+dEydOnGiF36PeHdxmJU+TAAAAAElFTkSuQmCC\" 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":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-06-19T03:22:07.000Z","updated_at":"2026-05-31T15:26:16.000Z","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,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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":18,"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":"2026-05-25T05:44:40.000Z","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":{"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":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-06-19T03:22:07.000Z","updated_at":"2026-05-31T15:26:16.000Z","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,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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":18,"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":"2026-05-25T05:44:40.000Z","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\"}]}"}],"errors":[],"facets":[[{"value":"Sequences \u0026 Series V","count":1,"selected":false}],[{"value":"medium","count":2,"selected":false}]],"term":"tag:\"totient\"","page":1,"per_page":50,"sort":"map(difficulty_value,0,0,999) asc"}}