DivisorSigma[0, n] == 2 Log[n, DivisorProduct[Subscript[d, j], {Subscript[d, j], n}]] /; Element[Subscript[d, j], Divisors[n]] && Element[n, Integers] && 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> <mn> 0 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msub> <mi> log </mi> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <munder> <mo> &#8719; </mo> <mrow> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> &#8739; </mo> <mi> n </mi> </mrow> </munder> <msub> <mi> d </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> &#8712; </mo> <mrow> <mi> divisors </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </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> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> &#963; </ms> <ci> DivisorSigma </ci> </apply> <ms> 0 </ms> </apply> <ms> ( </ms> <ms> n </ms> <ms> ) </ms> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> Log </ms> <ms> [ </ms> <ms> n </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> &#8719; </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> &#8739; </ms> <ms> n </ms> </list> </apply> </apply> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> </list> </apply> <ms> ] </ms> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> &#8712; </ms> <apply> <ci> RowBox </ci> <list> <ms> divisors </ms> <ms> ( </ms> <ms> n </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> &#8712; </ms> <apply> <ci> SuperscriptBox </ci> <ms> &#8469; </ms> <ms> + </ms> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DivisorSigma", "[", RowBox[List["0", ",", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["n", ",", RowBox[List["DivisorProduct", "[", RowBox[List[SubscriptBox["d", "j"], ",", RowBox[List["{", RowBox[List[SubscriptBox["d", "j"], ",", "n"]], "}"]]]], "]"]]]], "]"]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["d", "j"], "\[Element]", RowBox[List["Divisors", "[", "n", "]"]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

2001-10-29