Problem 57293. Compute the unitary totient of a number

Created by ChrisR

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The totient function </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"phi(n)\"/></w:customXmlPr><w:r><w:t>\\varphi(n)</w:t></w:r></w:customXml><w:r><w:t>, the subject of Cody Problems </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/656\"><w:r><w:rPr><w:u/></w:rPr><w:t>656</w:t></w:r></w:hyperlink><w:r><w:t> and </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/50182\"><w:r><w:rPr><w:u/></w:rPr><w:t>50182</w:t></w:r></w:hyperlink><w:r><w:rPr><w:u/></w:rPr><w:t>,</w:t></w:r><w:r><w:t> gives the number of integers smaller than </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> that are relatively prime to </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t>--that is, that share no common factors with </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> other than 1. Therefore, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"phi(9) = 6\"/></w:customXmlPr><w:r><w:t>\\varphi(9) = 6</w:t></w:r></w:customXml><w:r><w:t> because 1, 2, 4, 5, 7, and 8 (i.e., six numbers less than 9) are relatively prime to 9. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The unitary totient function </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"phi*(n)\"/></w:customXmlPr><w:r><w:t>\\varphi^*(n)</w:t></w:r></w:customXml><w:r><w:t> is defined in terms of the function gcd*(</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"k\"/></w:customXmlPr><w:r><w:t>k</w:t></w:r></w:customXml><w:r><w:t>,</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t>), which is the largest divisor of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"k\"/></w:customXmlPr><w:r><w:t>k</w:t></w:r></w:customXml><w:r><w:t> that is also a </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/56738-list-unitary-divisors-of-a-number\"><w:r><w:rPr><w:u/></w:rPr><w:t>unitary divisor</w:t></w:r></w:hyperlink><w:r><w:t> of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t>. Then the unitary totient function gives the number of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"k\"/></w:customXmlPr><w:r><w:t>k</w:t></w:r></w:customXml><w:r><w:t> (with </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"1 &lt;= k &lt;= n\"/></w:customXmlPr><w:r><w:t>1\\le k \\le n</w:t></w:r></w:customXml><w:r><w:t>) such that gcd*(</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"k\"/></w:customXmlPr><w:r><w:t>k</w:t></w:r></w:customXml><w:r><w:t>,</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t>) = 1. For example, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"phi*(9) = 8\"/></w:customXmlPr><w:r><w:t>\\varphi^*(9) = 8</w:t></w:r></w:customXml><w:r><w:t> because the unitary divisors of 9 are 1 and 9. Therefore, for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"1, 2, 3,...,8\"/></w:customXmlPr><w:r><w:t>k = 1, 2, 3,\\ldots,8</w:t></w:r></w:customXml><w:r><w:t>, the largest divisors that are also unitary divisors of 9 are 1-8. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function to compute the unitary totient function. </w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div 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; "><div style="block-size: 186px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 93px; transform-origin: 407px 93px; vertical-align: baseline; "><div 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; "><span style="block-size: auto; display: inline; margin-block-end: 0px; 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: 61.45px 7.79167px; transform-origin: 61.45px 7.79167px; unicode-bidi: normal; ">The totient function <img src="data:image/png;base64,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" alt="phi(n)" style="width: 31.5px; height: 18.5px;" width="31.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; 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: 97.2333px 7.79167px; transform-origin: 97.2333px 7.79167px; unicode-bidi: normal; ">, the subject of Cody Problems <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/656">656</a><span style="block-size: auto; display: inline; margin-block-end: 0px; 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; "> and <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/50182">50182</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; ">,<span style="block-size: auto; display: inline; margin-block-end: 0px; 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: 132.25px 7.79167px; transform-origin: 132.25px 7.79167px; unicode-bidi: normal; "> gives the number of integers smaller than n<span style="block-size: auto; display: inline; margin-block-end: 0px; 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: 15.55px 7.79167px; transform-origin: 15.55px 7.79167px; unicode-bidi: normal; "> that are relatively prime to n<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 136.117px 7.79167px; transform-origin: 136.117px 7.79167px; unicode-bidi: normal; ">--that is, that share no common factors with n<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 77.3917px 7.79167px; transform-origin: 77.3917px 7.79167px; unicode-bidi: normal; "> other than 1. Therefore, <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHMAAAAlCAYAAABxlNYMAAAFzUlEQVR4Xu2aN8tsVRSGv/sLjKVYGApBUDCBaKFgbgQVFUsFQyEI5lCIXLnXws4ECiKYQEEExVBYKKLcKygICoZCLM1/QN8H9oI1e3acOcGZbw68DHPCWnuvd6W9zzmwtzu2xgIHtmYmu4ns7cjcIifYkbkjcxALnCApJwnfDCJtb++sAWUNNKRpxcwVmRD5qvDAgAQcDKZ7ZFoTzqbtFGn+2Wufg8xWIhnsZcIxYcAfNhAPoccJd81m4nEUYzNsca1wrPCX8Irw/txkHtEAXhReyMybgT8jnCvcLfwjXCg8KPwk3CN8VrDZG7r2SUH+OOYeRyq2eEK4U/gzOOnH+v09pW7qyCRyzhGuLBD5pa6dKpwYDfoi/f80PHdxgVAi+qhwiTBUPR6HqrJU5vuuQKYhK92SI9HETEkmRiaySkR8oOtXCM8FL4yn+2zwUuRcUJjc/bp2g3DeHCwMoPMmyXg9yMnZYknNlGRCBITmopJu9Oswwpv1S7qMj6t14r1w8g795lK1OU5OzgD2Hk2Et0NTRE4dmS3GJZoOh4Gdrd9UiqSG/BbuIZWWIg9noFnIOc9obKwhmPlZmUFMzg5JFVNF5u3S/rxALVxop92oLMXWJvFveIaG4HQh2QzofIvONew+yqPeod+UBtJt89FDJgWZrtIvEfzy4RddS6VGBgNRpwXkBtdKJt0wnW6NdGuYNinV/hgcnrldI3wvnC+cLPwtfCFkm7oWMvHw+wQi4UbBIovO9GGBhT+KiDwak8vdPUYc0YQTlFIejoB8jhIBraQjB71PCr0bCTjp9Tb4NX4/17OlZZQX7bt1ztP4sCSJD8rLbcISqTUyrXuMQ97SAUQ+FbRZxMT3Wr2sGdXSok0kt/DvJbPmRCmuYsOuymepSYtl+hTLNWz7lkDwUEtvFaynIGiuiwktkZlbBhg5ROrxbkRm5LiWmWE88SnjmFyuIYNUmqqvf+g8ay+OWoOAg3H0LlGGisyWXSuzhdmb/znH9w6/VFNzZPqHyN1+28jSYbz+8RHjF/ytZDIJ750Ygug0QjHwvYKlnloDhDwbUy0DpZxr6nPefqW1uDnzksOnJklI/xC8P27/ffTECr0S32X2kBkTyn9I5aAROCRYqqktTTaZzFLG8b3FQqClyPTRETcidi2Oinih6xudXjIhAYdih+fMQCSNxHcCm822M1JL20ZmrYueOgJz+nyaLZHp+VmoySkyrT1OpbFck+PTclz0jegW49cM61v3eO829WxLF516bqia2dPNepLi0ubH2EymT6OlFBsT5h0g1bhg1O5FcGTlksPknAC9zXubTsgc3azPbqUuOLtTFkemn0TcUXlj+jTgN4Vzg1i1qzT7MlFea9kbhJYtOjNOz/LA9A0VmT3dLLotKErLKauZSy8bYjK9d8RkeiHUIatttpdYGoBtMKzSVVI/6fSI+J6NZ3Oy2vKlltqnvJ4LGD8GI3ypbKWMa11pnBZtT9SnLSvaNSObk5RqQcpoPiLR+5iQ24uNn8f5cABzvClJWUeX9SUpmxrZSXunyPRp04zvXz3R4fK2mzfg3IuRW7bL8KiPhJZPOkhz7HiwXUg64YuDhU8kKtay5dVDui/3mmwdg4/5rP+6ANJeFn4V2BdnWUbGfDrl1Lm0R+28SuCrAA4U2OY2wnj78Y6Q/YQhMVtzklIXSnG/VCCaIP41oXVv06tEF1mj9FZlTEKGkI1D2yY78r4V2HjPvXVq/gi6pTC3TIDa95WQi2ScaBXyvG57J/ioTube4rSMdePuaWlI/HJl3bXiFN/n0GyRObreBW4cc4kBt5DZ0mH12IKm5m2h9A1Pjzx/L7X9cYGlS2ujtKqu/91zLWTakiR+S7LOZEinpMHqF2cdSnASGoQhZXaon//WFjJtSbLuDk48W4x/xoB1jebppf0YkWbYGpl+J6R3N2N+V91nI6iRuc/MsdnT3ZG52fwtjH5H5o7MLbLAFk3lP3WAcjV1N9yEAAAAAElFTkSuQmCC" alt="phi(9) = 6" style="width: 57.5px; height: 18.5px;" width="57.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; 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: 61.4417px 7.79167px; transform-origin: 61.4417px 7.79167px; unicode-bidi: normal; "> because 1, 2, 4, 5, 7, and 8 (i.e., six numbers less than 9) are relatively prime to 9. </div><div 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; "><span style="block-size: auto; display: inline; margin-block-end: 0px; 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 7.79167px; transform-origin: 84.4px 7.79167px; unicode-bidi: normal; ">The unitary totient function <img src="data:image/png;base64,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" alt="phi*(n)" style="width: 36.5px; height: 19.5px;" width="36.5" height="19.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; 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: 122.908px 7.79167px; transform-origin: 122.908px 7.79167px; unicode-bidi: normal; "> is defined in terms of the function gcd*(k<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; ">,n<span style="block-size: auto; display: inline; margin-block-end: 0px; 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: 97.625px 7.79167px; transform-origin: 97.625px 7.79167px; unicode-bidi: normal; ">), which is the largest divisor of k<span style="block-size: auto; display: inline; margin-block-end: 0px; 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: 43.1667px 7.79167px; transform-origin: 43.1667px 7.79167px; unicode-bidi: normal; "> that is also a <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/56738-list-unitary-divisors-of-a-number">unitary divisor</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.71667px 7.79167px; transform-origin: 9.71667px 7.79167px; unicode-bidi: normal; "> of n<span style="block-size: auto; display: inline; margin-block-end: 0px; 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: 167.242px 7.79167px; transform-origin: 167.242px 7.79167px; unicode-bidi: normal; ">. Then the unitary totient function gives the number of k<span style="block-size: auto; display: inline; margin-block-end: 0px; 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: 18.6667px 7.79167px; transform-origin: 18.6667px 7.79167px; unicode-bidi: normal; "> (with <img src="data:image/png;base64,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" alt="1 <= k <= n" style="width: 62px; height: 18px;" width="62" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; 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: 50.95px 7.79167px; transform-origin: 50.95px 7.79167px; unicode-bidi: normal; ">) such that gcd*(k<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; ">,n<span style="block-size: auto; display: inline; margin-block-end: 0px; 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: 30.525px 7.79167px; transform-origin: 30.525px 7.79167px; unicode-bidi: normal; ">) = 1. For example, <img src="data:image/png;base64,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" alt="phi*(9) = 8" style="width: 63px; height: 19.5px;" width="63" height="19.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; 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: 187.858px 7.79167px; transform-origin: 187.858px 7.79167px; unicode-bidi: normal; "> because the unitary divisors of 9 are 1 and 9. Therefore, for <img src="data:image/png;base64,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" alt="1, 2, 3,...,8" style="width: 101px; height: 18px;" width="101" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 77.7833px 7.79167px; transform-origin: 77.7833px 7.79167px; unicode-bidi: normal; ">, the largest divisors that are also unitary divisors of 9 are 1-8. </div><div 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; "><span style="block-size: auto; display: inline; margin-block-end: 0px; 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: 171px 7.79167px; transform-origin: 171px 7.79167px; unicode-bidi: normal; ">Write a function to compute the unitary totient function. </div></div></div>

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Problem 57293. Compute the unitary totient of a number

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Problem 57293. Compute the unitary totient of a number

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