Problem 52809. Easy Sequences 28: Sum of Radicals of Integers

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 </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Radical_of_an_integer\"><w:r><w:t>radical of a positive integer</w:t></w:r></w:hyperlink><w:r><w:t> </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>x</w:t></w:r></w:customXml><w:r><w:t> is defined as the product of the distinct prime numbers dividing </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>x</w:t></w:r></w:customXml><w:r><w:t>. For example, the distinct prime factors of </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> is </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>[2\\ 5]</w:t></w:r></w:customXml><w:r><w:t>, therefore the radical of </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> is </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>10</w:t></w:r></w:customXml><w:r><w:t>. Similarly, the radicals of </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>14</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:customXmlPr><w:r><w:t>125</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>1344</w:t></w:r></w:customXml><w:r><w:t> are </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>14</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:customXmlPr><w:r><w:t>5</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>42</w:t></w:r></w:customXml><w:r><w:t>, respectively, 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>is considered to be the radical of itself.</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 the sum of the radicals of all positive 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:rPr><w:b/></w:rPr><w:t>.</w:t></w:r><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:t>For </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 = 10</w:t></w:r></w:customXml><w:r><w:t>, the radicals are: </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, 2, 5, 6, 7, 2, 3, 10]</w:t></w:r></w:customXml><w:r><w:t>. Therefore, </w:t></w:r><w:r><w:rPr><w:u/></w:rPr><w:t>the output should be </w:t></w:r><w:r><w:rPr><w:b/><w:u/></w:rPr><w:t>'41'</w:t></w:r><w:r><w:t>.</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: 123px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 61.5px; transform-origin: 407px 61.5px; 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: 14.5px 8px; transform-origin: 14.5px 8px; unicode-bidi: normal; ">The <a target='_blank' href = "https://en.wikipedia.org/wiki/Radical_of_an_integer">radical of a positive integer</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: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "> x<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: 201px 8px; transform-origin: 201px 8px; unicode-bidi: normal; "> is defined as the product of the distinct prime numbers dividing x<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: 59px 8px; transform-origin: 59px 8px; unicode-bidi: normal; ">. For example, the distinct prime factors of <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: 9px 8px; transform-origin: 9px 8px; unicode-bidi: normal; "> is <img src="data:image/png;base64,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" style="width: 35.5px; height: 18.5px;" width="35.5" 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: 78px 8px; transform-origin: 78px 8px; unicode-bidi: normal; ">, therefore the radical of <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: 9px 8px; transform-origin: 9px 8px; unicode-bidi: normal; "> is <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAkCAYAAADhAJiYAAABJUlEQVRYhe2WbRGEIBCGnw42oIAFTGACG9jABlYwgxHocBXMYAXvh3ByzqGwfowzxzvDD5nd5WGXFSApKelZUkAm8CnMOBWkAyYgD/SpgAHogcb4j0BN/KY+yhwQO0KArE+7mq/NvJZAlcw7K4FXBJBddPAsqj2wUWoCgTLmsmwtWAXGOgWoduxKj41ybMRZCgVyS6s27GwWh6uBrM24E08TBn4IKBcC+Up7GKgQAtV3APUR8W4B0jvx2ruBHlEyhECin2Mo0BAJNHJh28P3RRySyZcEJgbIPUeVx8b9X4nOTwwQLOXwdZrtMN9r4HQgxXJXrW3d14D4BalYDutk4PaUm4VHB0qxXL6i60Ixp1f/GB379c+MTW98evMt6qqkpKSkv9UbptKi8sNv99EAAAAASUVORK5CYII=" 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: 80px 8px; transform-origin: 80px 8px; unicode-bidi: normal; ">. Similarly, the radicals of <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAkCAYAAADhAJiYAAAA6klEQVRYhe2VWxGDMBBFj4c4wAAGogAFdRAHOMACGpCAByxUQy20Hwl0y0BLk5ZlhtyZ/UgmXM5uHgtZWVnHUgGYhO8NYBM9JpAWuANlgs+Q6mEEyBixZnWqRxVMKp6ZxZpZfpPUpJTsDHADuqMAdUDPa5XUgBxwxV8KdaASv1U2jNWBBqARY1WgJgDJB1ANyOK3ar5OBWi84m4FdHegLsSSdgdy+OpU4efzcMLjIuajGu0WoF6s+SaiqrUFqAlQayH74SDmi38BfZL6w3geoALfIEezWguoYP2gtiw/fu9UkniQs7Kysk6pB0ucjplAUbTSAAAAAElFTkSuQmCC" 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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, <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: 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: 14.5px 8px; transform-origin: 14.5px 8px; unicode-bidi: normal; "> are <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAkCAYAAADhAJiYAAAA6klEQVRYhe2VWxGDMBBFj4c4wAAGogAFdRAHOMACGpCAByxUQy20Hwl0y0BLk5ZlhtyZ/UgmXM5uHgtZWVnHUgGYhO8NYBM9JpAWuANlgs+Q6mEEyBixZnWqRxVMKp6ZxZpZfpPUpJTsDHADuqMAdUDPa5XUgBxwxV8KdaASv1U2jNWBBqARY1WgJgDJB1ANyOK3ar5OBWi84m4FdHegLsSSdgdy+OpU4efzcMLjIuajGu0WoF6s+SaiqrUFqAlQayH74SDmi38BfZL6w3geoALfIEezWguoYP2gtiw/fu9UkniQs7Kysk6pB0ucjplAUbTSAAAAAElFTkSuQmCC" 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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, 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: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "> and <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: 83.5px 8px; transform-origin: 83.5px 8px; unicode-bidi: normal; ">, respectively, The number1<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: 121.5px 8px; transform-origin: 121.5px 8px; unicode-bidi: normal; ">is considered to be the radical of itself.</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: 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: 182.5px 8px; transform-origin: 182.5px 8px; unicode-bidi: normal; ">, find the sum of the radicals of all positive integers <img src="data:image/png;base64,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" 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: 2px 8px; transform-origin: 2px 8px; 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: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "> </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: 13px 8px; transform-origin: 13px 8px; unicode-bidi: normal; ">For <img src="data:image/png;base64,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" style="width: 44px; height: 18px;" width="44" 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: 57px 8px; transform-origin: 57px 8px; unicode-bidi: normal; ">, the radicals are: <img src="data:image/png;base64,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" style="width: 152.5px; height: 18.5px;" width="152.5" 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: 39.5px 8px; transform-origin: 39.5px 8px; unicode-bidi: normal; ">. Therefore, <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: 67px 8px; transform-origin: 67px 8px; unicode-bidi: normal; ">the output should be <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: 11px 8px; transform-origin: 11px 8px; unicode-bidi: normal; ">'41'<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: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; ">.</div></div></div>

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Problem 52809. Easy Sequences 28: Sum of Radicals of Integers

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Problem 52809. Easy Sequences 28: Sum of Radicals of Integers

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