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FactorInteger






Mathematica Notation

Traditional Notation









Number Theory Functions > FactorInteger[n] > Identities > Functional identities





http://functions.wolfram.com/13.01.17.0001.01









  


  










Input Form





x^(\[Omega][n m] - \[Omega][m]) == DivisorSum[(x^\[Omega][n/d] (1 - t))^ \[CapitalOmega][d], {d, GCD[n, m]}] /; Element[x, Reals] && (\[Omega][k] == r /; k == Product[Prime[j]^Subscript[n, j], {j, 1, r}]) && (\[CapitalOmega][k] == Sum[Subscript[n, j], {j, 1, r}] /; k == Product[Prime[j]^Subscript[n, j], {j, 1, r}])










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> x </mi> <mrow> <mrow> <mi> &#969; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> &#969; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#10869; </mo> <mrow> <munder> <mo> &#8721; </mo> <mrow> <mi> d </mi> <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> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mrow> <mi> &#969; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> n </mi> <mi> d </mi> </mfrac> <mo> ) </mo> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#937; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> &#969; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mi> r </mi> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </munderover> <msup> <mrow> <mi> prime </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> j </mi> <mo> ) </mo> </mrow> <msub> <mi> n </mi> <mi> j </mi> </msub> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> &#937; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </munderover> <msub> <mi> n </mi> <mi> j </mi> </msub> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </munderover> <msup> <mrow> <mi> prime </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> j </mi> <mo> ) </mo> </mrow> <msub> <mi> n </mi> <mi> j </mi> </msub> </msup> </mrow> </mrow> </mrow> <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> SuperscriptBox </ci> <ms> x </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#969; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> m </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> &#969; </ms> <ms> ( </ms> <ms> m </ms> <ms> ) </ms> </list> </apply> </list> </apply> </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> <ms> d </ms> <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> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> x </ms> <apply> <ci> RowBox </ci> <list> <ms> &#969; </ms> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> n </ms> <ms> d </ms> </apply> <ms> ) </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> t </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#937; </ms> <ms> ( </ms> <ms> d </ms> <ms> ) </ms> </list> </apply> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> x </ms> <ms> &#8712; </ms> <apply> <ci> TagBox </ci> <ms> &#8477; </ms> <apply> <ci> Function </ci> <reals /> </apply> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#969; </ms> <ms> ( </ms> <ms> k </ms> <ms> ) </ms> </list> </apply> <ms> &#10869; </ms> <ms> r </ms> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> r </ms> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> prime </ms> <ms> ( </ms> <ms> j </ms> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> j </ms> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#937; </ms> <ms> ( </ms> <ms> k </ms> <ms> ) </ms> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> r </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> j </ms> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> r </ms> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> prime </ms> <ms> ( </ms> <ms> j </ms> <ms> ) </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> j </ms> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> <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", "[", SuperscriptBox["x_", RowBox[List[RowBox[List["\[Omega]", "[", RowBox[List["n_", " ", "m_"]], "]"]], "-", RowBox[List["\[Omega]", "[", "m_", "]"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["DivisorSum", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["x", RowBox[List["\[Omega]", "[", FractionBox["n", "d"], "]"]]], " ", RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]]]], ")"]], RowBox[List["\[CapitalOmega]", "[", "d", "]"]]], ",", RowBox[List["{", RowBox[List["d", ",", RowBox[List["GCD", "[", RowBox[List["n", ",", "m"]], "]"]]]], "}"]]]], "]"]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["\[Omega]", "[", "k", "]"]], "\[Equal]", "r"]], "/;", RowBox[List["k", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "r"], SuperscriptBox[RowBox[List["Prime", "[", "j", "]"]], SubscriptBox["n", "j"]]]]]]]], ")"]], "&&", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["\[CapitalOmega]", "[", "k", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "r"], SubscriptBox["n", "j"]]]]], "/;", RowBox[List["k", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "r"], SuperscriptBox[RowBox[List["Prime", "[", "j", "]"]], SubscriptBox["n", "j"]]]]]]]], ")"]]]]]]]]]]










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





2001-10-29





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