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

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["DivisorSigma", "[", RowBox[List["k", ",", "n"]], "]"]], "\[Equal]", RowBox[List["Product", "[", RowBox[List[RowBox[List["DivisorSigma", "[", RowBox[List["k", ",", SubsuperscriptBox["p", "j", SubscriptBox["n", "j"]]]], "]"]], ",", RowBox[List["{", RowBox[List["j", ",", "1", ",", "m"]], "}"]]]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["FactorInteger", "[", "n", "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["p", "1"], ",", SubscriptBox["n", "1"]]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["p", "m"], ",", SubscriptBox["n", "m"]]], "}"]]]], "}"]]]], "\[And]", RowBox[List[SubscriptBox["p", "j"], "\[Element]", "Primes"]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

<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> <mrow> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, DivisorSigma] </annotation> </semantics> <mi> k </mi> </msub> <mo> ( </mo> <msubsup> <mi> p </mi> <mi> j </mi> <msub> <mi> n </mi> <mi> j </mi> </msub> </msubsup> <mo> ) </mo> </mrow> </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> j </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> k </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </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> <ci> DivisorSigma </ci> <ci> k </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> j </ci> </apply> </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> j </ci> </apply> <primes /> </apply> <apply> <in /> <ci> k </ci> <integers /> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DivisorSigma", "[", RowBox[List["k", ",", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "m"], RowBox[List["DivisorSigma", "[", RowBox[List["k", ",", SubsuperscriptBox["p", "j", SubscriptBox["n", "j"]]]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["FactorInteger", "[", "n", "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["p", "1"], ",", SubscriptBox["nn", "1"]]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["p", "m"], ",", SubscriptBox["n", "m"]]], "}"]]]], "}"]]]], "&&", RowBox[List[SubscriptBox["p", "j"], "\[Element]", "Primes"]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

2001-10-29