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DivisorSigma




Mathematica Notation

Traditional Notation









Number Theory Functions > DivisorSigma[k,n] > Identities > Functional identities





http://functions.wolfram.com/13.05.17.0001.01









  


  








Input Form





DivisorSum[DivisorSigma[0, Subscript[d, j]]^3, {Subscript[d, j], n}] == DivisorSum[DivisorSigma[0, Subscript[d, j]], {Subscript[d, j], n}]^2 /; Element[Subscript[d, j], Divisors[n]]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["DivisorSum", "[", RowBox[List[SuperscriptBox[RowBox[List["DivisorSigma", "[", RowBox[List["0", ",", SubscriptBox["d", "j"]]], "]"]], "3"], ",", RowBox[List["{", RowBox[List[SubscriptBox["d", "j"], ",", "n"]], "}"]]]], "]"]], "\[Equal]", SuperscriptBox[RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List["DivisorSigma", "[", RowBox[List["0", ",", SubscriptBox["d", "j"]]], "]"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["d", "j"], ",", "n"]], "}"]]]], "]"]], "2"]]], "/;", RowBox[List[SubscriptBox["d", "j"], "\[Element]", RowBox[List["Divisors", "[", "n", "]"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munder> <mo> &#8721; </mo> <mrow> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> | </mo> <mi> n </mi> </mrow> </munder> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mi> &#963; </mi> <mn> 0 </mn> </msub> <mo> ( </mo> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> &#10869; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <munder> <mo> &#8721; </mo> <mrow> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> | </mo> <mi> n </mi> </mrow> </munder> <mrow> <msub> <mi> &#963; </mi> <mn> 0 </mn> </msub> <mo> ( </mo> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> /; </mo> <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> </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> UnderscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> &#8721; </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> | </ms> <ms> n </ms> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#963; </ms> <ms> 0 </ms> </apply> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> ) </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 3 </ms> </apply> </list> </apply> <ms> &#10869; </ms> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> | </ms> <ms> n </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#963; </ms> <ms> 0 </ms> </apply> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> /; </ms> <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> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DivisorSum", "[", RowBox[List[SuperscriptBox[RowBox[List["DivisorSigma", "[", RowBox[List["0", ",", SubscriptBox["d_", "j"]]], "]"]], "3"], ",", RowBox[List["{", RowBox[List[SubscriptBox["d_", "j"], ",", "n_"]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox[RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List["DivisorSigma", "[", RowBox[List["0", ",", SubscriptBox["d", "j"]]], "]"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["d", "j"], ",", "n"]], "}"]]]], "]"]], "2"], "/;", RowBox[List[SubscriptBox["d", "j"], "\[Element]", RowBox[List["Divisors", "[", "n", "]"]]]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2001-10-29