DivisorSigma[k, n] == Product[(Subscript[p, j]^((Subscript[n, j] + 1) k) - 1)/ (Subscript[p, j]^k - 1), {j, 1, m}] /; FactorInteger[n] == {{Subscript[p, 1], Subscript[n, 1]}, \[Ellipsis], {Subscript[p, m], Subscript[n, m]}} && Element[Subscript[p, k], Primes] && n > 0

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, DivisorSigma] </annotation> </semantics> <mi> k </mi> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mfrac> <mrow> <msubsup> <mi> p </mi> <mi> j </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mi> j </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <msubsup> <mi> p </mi> <mi> j </mi> <mi> k </mi> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> factors </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> p </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> p </mi> <mi> m </mi> </msub> <mo> , </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> p </mi> <mi> k </mi> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8473; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalP]&quot;, Function[Primes]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> DivisorSigma </ci> <ci> k </ci> <ci> n </ci> </apply> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> factors </ci> <ci> n </ci> </apply> <list> <list> <apply> <ci> Subscript </ci> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </list> <ci> &#8230; </ci> <list> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> m </ci> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> m </ci> </apply> </list> </list> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> <primes /> </apply> <apply> <gt /> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29