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DivisorSigma




Mathematica Notation

Traditional Notation









Number Theory Functions > DivisorSigma[k,n] > Transformations > Products, sums, and powers of the direct function > Products of the direct function





http://functions.wolfram.com/13.05.16.0002.01









  


  








Input Form





DivisorSigma[k, m] DivisorSigma[k, n] == DivisorSum[Subscript[d, j]^k DivisorSigma[k, (m n)/Subscript[d, j]^2], {Subscript[d, j], GCD[n, m]}] /; Element[Subscript[d, j], Divisors[GCD[n, m]]]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["DivisorSigma", "[", RowBox[List["k", ",", "m"]], "]"]], RowBox[List["DivisorSigma", "[", RowBox[List["k", ",", "n"]], "]"]]]], "\[Equal]", RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List[SubsuperscriptBox["d", "j", "k"], " ", RowBox[List["DivisorSigma", "[", RowBox[List["k", ",", FractionBox[RowBox[List["m", " ", "n"]], SubsuperscriptBox["d", "j", "2"]]]], "]"]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["d", "j"], ",", RowBox[List["GCD", "[", RowBox[List["n", ",", "m"]], "]"]]]], "}"]]]], "]"]]]], "/;", RowBox[List[SubscriptBox["d", "j"], "\[Element]", RowBox[List["Divisors", "[", RowBox[List["GCD", "[", RowBox[List["n", ",", "m"]], "]"]], "]"]]]]]]]]








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> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, DivisorSigma] </annotation> </semantics> <mi> k </mi> </msub> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <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> </mrow> <mo> &#10869; </mo> <mrow> <mstyle scriptlevel='0'> <munder> <mo> &#8721; </mo> <mrow> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> | </mo> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </munder> </mstyle> <mstyle scriptlevel='0'> <mtext> </mtext> </mstyle> <mrow> <mstyle scriptlevel='0'> <msubsup> <mi> d </mi> <mi> j </mi> <mi> k </mi> </msubsup> </mstyle> <mo> &#8290; </mo> <mstyle scriptlevel='0'> <mtext> </mtext> </mstyle> <mrow> <mstyle scriptlevel='0'> <msub> <mi> &#963; </mi> <mrow> <mtext> </mtext> <mi> k </mi> </mrow> </msub> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mtext> </mtext> </mstyle> <mstyle scriptlevel='0'> <mfrac> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mrow> <mtext> </mtext> <msubsup> <mi> d </mi> <mi> j </mi> <mn> 2 </mn> </msubsup> </mrow> </mfrac> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </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> k </ms> </apply> <ms> ( </ms> <ms> m </ms> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> &#963; </ms> <ci> DivisorSigma </ci> </apply> <ms> k </ms> </apply> <ms> ( </ms> <ms> n </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> &#10869; </ms> <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> <apply> <ci> RowBox </ci> <list> <ms> gcd </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> , </ms> <ms> m </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> d </ms> <ms> j </ms> <ms> k </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#963; </ms> <ms> k </ms> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> n </ms> </list> </apply> <apply> <ci> SubsuperscriptBox </ci> <ms> d </ms> <ms> j </ms> <ms> 2 </ms> </apply> </apply> <ms> ) </ms> </list> </apply> </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[RowBox[List["DivisorSigma", "[", RowBox[List["k", ",", "m_"]], "]"]], " ", RowBox[List["DivisorSigma", "[", RowBox[List["k", ",", "n_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List[SubsuperscriptBox["d", "j", "k"], " ", RowBox[List["DivisorSigma", "[", RowBox[List["k", ",", FractionBox[RowBox[List["m", " ", "n"]], SubsuperscriptBox["d", "j", "2"]]]], "]"]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["d", "j"], ",", RowBox[List["GCD", "[", RowBox[List["n", ",", "m"]], "]"]]]], "}"]]]], "]"]], "/;", RowBox[List[SubscriptBox["d", "j"], "\[Element]", RowBox[List["Divisors", "[", RowBox[List["GCD", "[", RowBox[List["n", ",", "m"]], "]"]], "]"]]]]]]]]]]


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





2001-10-29