Problem 52804. Easy Sequences 29: Odd proper divisors of odd proper divisors

Created by Ramon Villamangca

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 number </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>210</w:t></w:r></w:customXml><w:r><w:t> is special. It has odd number of proper divisors: </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>[1,2,3,5,6,7,10,14,15,21,30,35,42,70,105]</w:t></w:r></w:customXml><w:r><w:t>. Furthermore, if you take </w:t></w:r><w:r><w:rPr><w:u/></w:rPr><w:t>any</w:t></w:r><w:r><w:t> of its proper divisors, say </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>70</w:t></w:r></w:customXml><w:r><w:t>, it too has odd number of proper divisors: </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>[1,2,5,7,10,14,35]</w:t></w:r></w:customXml><w:r><w:t>. The numbers </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>22</w:t></w:r></w:customXml><w:r><w:t> and </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>5187</w:t></w:r></w:customXml><w:r><w:t>, have similar property as </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>210</w:t></w:r></w:customXml><w:r><w:t>. </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:rPr><w:b/></w:rPr><w:t>Given a limit </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:rPr><w:b/></w:rPr><w:t>, find how many integers </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>\\leq n</w:t></w:r></w:customXml><w:r><w:t>, have similar property as 210, namely, </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>the integers should have odd number of proper divisors and </w:t></w:r><w:r><w:rPr><w:b/><w:u/></w:rPr><w:t>all</w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t> its proper divisors have odd number of proper divisors</w:t></w:r><w:r><w:rPr/><w:t>, as well. </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 number </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>100</w:t></w:r></w:customXml><w:r><w:t>, does not qualify because it has even proper divisors, </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>8</w:t></w:r></w:customXml><w:r><w:t> in total </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>[1,2,4,5,10,20,25,50]</w:t></w:r></w:customXml><w:r><w:t>. The number </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>108</w:t></w:r></w:customXml><w:r><w:t> also doesn't qualify because although it has </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>11</w:t></w:r></w:customXml><w:r><w:t>proper divisors, some of its divisor, like </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>36</w:t></w:r></w:customXml><w:r><w:t>, have even number of proper divisors. </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:rPr><w:b/></w:rPr><w:t>NOTE:</w:t></w:r><w:r><w:t> A proper divisor of a number, is a divisor which is less than the number. Exception to this rule is the number </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>1</w:t></w:r></w:customXml><w:r><w:t>, which is considered a proper divisor of itself.</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: 237px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 118.5px; transform-origin: 407px 118.5px; 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: 40.5px 8px; transform-origin: 40.5px 8px; unicode-bidi: normal; ">The number <img src="data:image/png;base64,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" style="width: 25.5px; height: 18px;" width="25.5" 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: 155px 8px; transform-origin: 155px 8px; unicode-bidi: normal; "> is special. It has odd number of proper divisors: <img src="data:image/png;base64,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" style="width: 288.5px; height: 18.5px;" width="288.5" 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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">. Furthermore, if you take <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: 11.5px 8px; transform-origin: 11.5px 8px; unicode-bidi: normal; ">any<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: 83px 8px; transform-origin: 83px 8px; unicode-bidi: normal; "> of its proper divisors, say <img src="data:image/png;base64,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" style="width: 18px; height: 18px;" width="18" 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: 134px 8px; transform-origin: 134px 8px; unicode-bidi: normal; ">, it too has odd number of proper divisors: <img src="data:image/png;base64,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" style="width: 126px; height: 18.5px;" width="126" 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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">. The numbers <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAkCAYAAADhAJiYAAAA+klEQVRYhe2WTQ2EMBBGn4c6wAAGVgEKcICDdbAW0FAJeMBCNWBh99CS8NPpEqYQDn3JXAjvS5uZQqFQKDyDV6j6ZneFAXrgu6kJ6C50o1RB3gYu63OBKzIADmgXz2r2u24yu1Ea/A6lnr8XoTajK2JJ99mwnolcrkiPn4MUgxCqcVXMocPNrogLoWeOsMaNUoVAx//25HRF5pNyZocaN4rBD+OZ/mtcEQuMIfxON0qH7/2Zn6TGjdIqAjXu8xfTHAisiR9hjSu+POEvVxIGP6jbUI2bXMyI/3ZI5dgfY42bXEzqgiXdazSuiA0rP1KW9XdF4xYKhUIhxg+zAZ+7e7/gtwAAAABJRU5ErkJggg==" style="width: 18px; height: 18px;" width="18" 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: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "> and <img src="data:image/png;base64,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" style="width: 33px; height: 18px;" width="33" 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: 81.5px 8px; transform-origin: 81.5px 8px; unicode-bidi: normal; ">, have similar property as <img src="data:image/png;base64,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" style="width: 25.5px; height: 18px;" width="25.5" 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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">. </div><div 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; "><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: 45.5px 8px; transform-origin: 45.5px 8px; unicode-bidi: normal; ">Given a limit 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: 87.5px 8px; transform-origin: 87.5px 8px; unicode-bidi: normal; ">, find how many integers <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADIAAAAkCAYAAADLsGk3AAABXElEQVRYhe2XbRGDMAyGHw84wEANVMEUzMEc4GAW0ICEecACGrDAfrS5ZT3YoJSP7frc9W5Hdk3eNEkBMplMJpM5FwVwOTqINRRABfR+/RxawODX/dCIFjIloDwyqCX8vIASqNlHQDmyrwEsLpHRm9a8gt9CwM3v+VA+jLdZoFPPWxaK2UOAYHAjW/x0/vkVVwFXoFH22eM9FFCzfQ8Uyl+FC7bjlf1K2WfHYnk/5h53/NH1OQN9IhfvUwcsJ9LGbB6W15aC7spHiysnjfRJtcbJHoJ0Q9eBzSibTeFsStDa/il578kwQTflLymp7xQJdPC/Q6Q/msj9v5LqltejdaxcxRb2TXJCQUtLQAIdy7ieZqX3FTW5liCCum9/VFg+Z1x6Um70lh2/dwrmTzQZu1NlJWXX4xI01kOnQN6xpr5hjLc3RIxem2CdgmHlOs2n7mPl2mzeZzKZzH/wBNgaraIIzz2QAAAAAElFTkSuQmCC" style="width: 25px; height: 18px;" width="25" 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: 123.5px 8px; transform-origin: 123.5px 8px; unicode-bidi: normal; ">, have similar property as 210, namely, <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: 106px 8px; transform-origin: 106px 8px; unicode-bidi: normal; ">the integers should have odd number of proper divisors and <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: 8px 8px; transform-origin: 8px 8px; unicode-bidi: normal; ">all<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: 196.5px 8px; transform-origin: 196.5px 8px; unicode-bidi: normal; "> its proper divisors have odd number of proper divisors<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: 29px 8px; transform-origin: 29px 8px; unicode-bidi: normal; ">, as well. </div><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: 40.5px 8px; transform-origin: 40.5px 8px; unicode-bidi: normal; ">The number <img src="data:image/png;base64,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" style="width: 25.5px; height: 18px;" width="25.5" 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: 172.5px 8px; transform-origin: 172.5px 8px; unicode-bidi: normal; ">, does not qualify because it has even proper divisors, 8<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: 27px 8px; transform-origin: 27px 8px; unicode-bidi: normal; "> in total <img src="data:image/png;base64,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" style="width: 147.5px; height: 18.5px;" width="147.5" 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: 44.5px 8px; transform-origin: 44.5px 8px; unicode-bidi: normal; ">. The number <img src="data:image/png;base64,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" style="width: 25.5px; height: 18px;" width="25.5" 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: 142px 8px; transform-origin: 142px 8px; unicode-bidi: normal; "> also doesn't qualify because although it has <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAkCAYAAADhAJiYAAAAdklEQVRYhe2OwQmAMBAEp4d0kAbsw0bsIB3YQmqxB1uwF/0YCHiPyGEU3IHjXrMMCCG+RQTCC645loEdGDq6F0I1Vq511OOajEA6/3pz1OM2kRyjHldBClKQghSkIAX9MigCWzWaOrnm2AwsxmVgesgVQghRcwBVnXRb64UZ7QAAAABJRU5ErkJggg==" style="width: 18px; height: 18px;" width="18" 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: 123.5px 8px; transform-origin: 123.5px 8px; unicode-bidi: normal; ">proper divisors, some of its divisor, like <img src="data:image/png;base64,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" style="width: 18px; height: 18px;" width="18" 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: 73px 8px; transform-origin: 73px 8px; unicode-bidi: normal; ">, have even number of proper divisors. </div><div 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; "><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: 21px 8px; transform-origin: 21px 8px; unicode-bidi: normal; ">NOTE:<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: 340px 8px; transform-origin: 340px 8px; unicode-bidi: normal; "> A proper divisor of a number, is a divisor which is less than the number. Exception to this rule is the number 1<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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, which is considered a proper divisor of itself.</div></div></div>

Solve

Solution Stats

4 Solutions
3 Solvers

Last Solution submitted on Jan 05, 2022

Last 200 Solutions

Problem Comments

Problem Recent Solvers3

Problem Tags

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Problem 52804. Easy Sequences 29: Odd proper divisors of odd proper divisors

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers3

Suggested Problems

More from this Author66

Problem Tags

Community Treasure Hunt

Problem 52804. Easy Sequences 29: Odd proper divisors of odd proper divisors

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers3

Suggested Problems

More from this Author66

Problem Tags

Community Treasure Hunt

Players