Problem 57417. Easy Sequences 86: The "real" Hyperprimorials

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>In </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/57412-easy-sequences-85-hyperprimorials\"><w:r><w:t>Easy Sequences 85</w:t></w:r></w:hyperlink><w:r><w:t>, we define the hyperprimorial 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>n</w:t></w:r></w:customXml><w:r><w:t>, as a exponent ladder of primes numbers from the </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>-th prime down to first prime (</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</w:t></w:r></w:customXml><w:r><w:t>). OEIS, however, defines hyperprimorial differently. In OEIS sequence </w:t></w:r><w:hyperlink w:docLocation=\"https://oeis.org/A076265\"><w:r><w:t>#A076265,</w:t></w:r></w:hyperlink><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>the hyperprimorial 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>n</w:t></w:r></w:customXml><w:r><w:t>,</w:t></w:r><w:r><w:rPr><w:b/></w:rPr><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>H_n</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>)  defined as follows:</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>                             </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>H_n=\\prod_{i=1}^n p_i^{p_{i}}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                             where: </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>p_i</w:t></w:r></w:customXml><w:r><w:t> is the </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>i</w:t></w:r></w:customXml><w:r><w:t>-th prime number.</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>Using the above formula, we want to calculate </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>S_n</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, which is the sum of all hyperprimorials from 1 to n:</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>                            </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>S_n = \\sum_{i=1}^{n} H_n</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>For example:</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>                            </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>S_5 = \\sum_{i=1}^{5} H_5=\\prod_{i=1}^1 p_i^{p_{i}}+\\prod_{i=1}^2 p_i^{p_{i}}+\\prod_{i=1}^3 p_i^{p_{i}}+\\prod_{i=1}^4 p_i^{p_{i}}+\\prod_{i=1}^5 p_i^{p_{i}}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                                </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>=2^2+(2^2\\cdot3^3)+(2^2\\cdot3^3\\cdot5^5)+(2^2\\cdot3^3\\cdot5^5\\cdot7^7)+(2^2\\cdot3^3\\cdot5^5\\cdot7^7\\cdot11^{11})</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><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>=79301169838401181987612</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:t>Since this will involve large integers, </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>please output a vector containing first 6, last 6, and the number of digits 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>S_n</w:t></w:r></w:customXml><w:r><w:t>. Therefore, the final function output should be:  </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>S_n=[793011 \\ 987612 \\ 23] </w:t></w:r></w:customXml><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: 430px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 215px; transform-origin: 407px 215px; vertical-align: baseline; "><div style="block-size: 64px; 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 32px; text-align: left; transform-origin: 384px 32px; 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: 7.5px 8px; transform-origin: 7.5px 8px; unicode-bidi: normal; "><span style="">In </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/57412-easy-sequences-85-hyperprimorials"><span style=""><span style="">Easy Sequences 85</span></span></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: 103px 8px; transform-origin: 103px 8px; unicode-bidi: normal; "><span style="">, we define the hyperprimorial of </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</span><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: 161px 8px; transform-origin: 161px 8px; unicode-bidi: normal; "><span style="">, as a exponent ladder of primes numbers from the </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</span><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: 30px 8px; transform-origin: 30px 8px; unicode-bidi: normal; "><span style="">-th prime down to first prime (</span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);">2</span><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: 221.5px 8px; transform-origin: 221.5px 8px; unicode-bidi: normal; "><span style="">). OEIS, however, defines hyperprimorial differently. In OEIS sequence </span></span><a target='_blank' href = "https://oeis.org/A076265"><span style=""><span style="">#A076265,</span></span></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; "><span style=""> </span></span><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: 13.5px 8px; transform-origin: 13.5px 8px; unicode-bidi: normal; "><span style="font-weight: 700; ">the hyperprimorial of </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</span><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="">,</span></span><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: 4.5px 8px; transform-origin: 4.5px 8px; unicode-bidi: normal; "><span style="font-weight: 700; "> (</span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" style="width: 19px; height: 20px;" width="19" height="20"></span><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: 72.5px 8px; transform-origin: 72.5px 8px; unicode-bidi: normal; "><span style="font-weight: 700; ">)  defined as follows:</span></span></div><div style="block-size: 45px; 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 22.5px; text-align: left; transform-origin: 384px 22.5px; 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: 58px 8px; transform-origin: 58px 8px; unicode-bidi: normal; "><span style="">                             </span></span><span style="vertical-align:-17px"><img src="data:image/png;base64,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" style="width: 76px; height: 45px;" width="76" height="45"></span></div><div style="block-size: 22px; 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 11px; text-align: left; transform-origin: 384px 11px; 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: 81px 8px; transform-origin: 81px 8px; unicode-bidi: normal; "><span style="">                             where: </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" style="width: 13.5px; height: 20px;" width="13.5" height="20"></span><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; "><span style=""> is the </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">i</span><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: 55.5px 8px; transform-origin: 55.5px 8px; unicode-bidi: normal; "><span style="">-th prime number.</span></span></div><div style="block-size: 22px; 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 11px; text-align: left; transform-origin: 384px 11px; 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: 163px 8px; transform-origin: 163px 8px; unicode-bidi: normal; "><span style="font-weight: 700; ">Using the above formula, we want to calculate </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" style="width: 15.5px; height: 20px;" width="15.5" height="20"></span><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: 180px 8px; transform-origin: 180px 8px; unicode-bidi: normal; "><span style="font-weight: 700; ">, which is the sum of all hyperprimorials from 1 to n:</span></span></div><div style="block-size: 45px; 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 22.5px; text-align: left; transform-origin: 384px 22.5px; 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: 56px 8px; transform-origin: 56px 8px; unicode-bidi: normal; "><span style="">                            </span></span><span style="vertical-align:-17px"><img src="data:image/png;base64,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" style="width: 74px; height: 45px;" width="74" height="45"></span></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: 41px 8px; transform-origin: 41px 8px; unicode-bidi: normal; "><span style="">For example:</span></span></div><div style="block-size: 45px; 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 22.5px; text-align: left; transform-origin: 384px 22.5px; 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: 56px 8px; transform-origin: 56px 8px; unicode-bidi: normal; "><span style="">                            </span></span><span style="vertical-align:-17px"><img src="data:image/png;base64,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" style="width: 353px; height: 45px;" width="353" height="45"></span></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: 64px 8px; transform-origin: 64px 8px; unicode-bidi: normal; "><span style="">                                </span></span><span style="vertical-align:-5px"><img 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" style="width: 441.5px; height: 19.5px;" width="441.5" height="19.5"></span></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: 64px 8px; transform-origin: 64px 8px; unicode-bidi: normal; "><span style="">                                </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" style="width: 190px; height: 18px;" width="190" height="18"></span><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="">.</span></span></div><div style="block-size: 43px; 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 21.5px; text-align: left; transform-origin: 384px 21.5px; 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: 115px 8px; transform-origin: 115px 8px; unicode-bidi: normal; "><span style="">Since this will involve large integers, </span></span><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: 264px 8px; transform-origin: 264px 8px; unicode-bidi: normal; "><span style="font-weight: 700; ">please output a vector containing first 6, last 6, and the number of digits of </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" style="width: 15.5px; height: 20px;" width="15.5" height="20"></span><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: 152.5px 8px; transform-origin: 152.5px 8px; unicode-bidi: normal; "><span style="">. Therefore, the final function output should be:  </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" style="width: 163.5px; height: 20px;" width="163.5" height="20"></span><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="">.</span></span></div></div></div>

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Problem 57417. Easy Sequences 86: The "real" Hyperprimorials

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Problem 57417. Easy Sequences 86: The "real" Hyperprimorials

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