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GCD






Mathematica Notation

Traditional Notation









Integer Functions > GCD[n1,n2,...,nm] > Summation > Finite summation





http://functions.wolfram.com/04.08.23.0001.01









  


  










Input Form





Sum[\[Ellipsis] Sum[F[GCD[Subscript[k, 1], Subscript[k, 2], \[Ellipsis], Subscript[k, m]]], {Subscript[k, m], 1, n}], {Subscript[k, 1], 1, n}, {Subscript[k, 2], 1, n}] == Sum[f[d] Floor[n/d]^m, {k, 1, n}] /; F[n] == DivisorSum[f[d], {d, n}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[StyleBox[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "1"]], "n"], Rule[SpanMinSize, 3]], RowBox[List[StyleBox[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "1"]], "n"], Rule[SpanMinSize, 3]], RowBox[List[StyleBox["\[Ellipsis]", Rule[SpanMinSize, 3]], RowBox[List[StyleBox[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "m"], "=", "1"]], "n"], Rule[SpanMinSize, 3]], RowBox[List["F", "[", RowBox[List["GCD", "[", RowBox[List[SubscriptBox["k", "1"], ",", SubscriptBox["k", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["k", "m"]]], "]"]], "]"]]]]]]]]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[RowBox[List["f", "[", "d", "]"]], SuperscriptBox[RowBox[List["\[LeftFloor]", FractionBox["n", "d"], "\[RightFloor]"]], "m"]]]]]]], "/;", RowBox[List[RowBox[List["F", "[", "n", "]"]], "\[Equal]", RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List["f", "[", "d", "]"]], ",", RowBox[List["{", RowBox[List["d", ",", "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> <mstyle minsize='3'> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> </mstyle> <mrow> <mstyle minsize='3'> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> </mstyle> <mrow> <mo minsize='3'> &#8230; </mo> <mo> &#8290; </mo> <mstyle minsize='3'> <mtext> </mtext> </mstyle> <mrow> <mstyle minsize='3'> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mi> m </mi> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> </mstyle> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> k </mi> <mi> m </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> &#8970; </mo> <mfrac> <mi> n </mi> <mi> d </mi> </mfrac> <mo> &#8971; </mo> </mrow> <mi> m </mi> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munder> <mo> &#8721; </mo> <mrow> <mi> d </mi> <mo> | </mo> <mi> n </mi> </mrow> </munder> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </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> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#8230; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> m </ms> </apply> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> F </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> gcd </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> m </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> f </ms> <ms> ( </ms> <ms> d </ms> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#8970; </ms> <apply> <ci> FractionBox </ci> <ms> n </ms> <ms> d </ms> </apply> <ms> &#8971; </ms> </list> </apply> <ms> m </ms> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> F </ms> <ms> ( </ms> <ms> n </ms> <ms> ) </ms> </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> <ms> d </ms> <ms> | </ms> <ms> n </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> f </ms> <ms> ( </ms> <ms> d </ms> <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[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k_", "1"], "=", "1"]], "n_"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k_", "2"], "=", "1"]], "n_"], RowBox[List["\[Ellipsis]_", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k_", "m_"], "=", "1"]], "n_"], RowBox[List["F", "[", RowBox[List["GCD", "[", RowBox[List[SubscriptBox["k_", "1"], ",", SubscriptBox["k_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["k_", "m_"]]], "]"]], "]"]]]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[RowBox[List["f", "[", "d", "]"]], " ", SuperscriptBox[RowBox[List["Floor", "[", FractionBox["n", "d"], "]"]], "m"]]]]], "/;", RowBox[List[RowBox[List["F", "[", "n", "]"]], "\[Equal]", RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List["f", "[", "d", "]"]], ",", RowBox[List["{", RowBox[List["d", ",", "n"]], "}"]]]], "]"]]]]]]]]]]










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





2001-10-29





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