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http://functions.wolfram.com/04.08.23.0004.01
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Sum[KroneckerDelta[1, GCD[k, n]]/(k n (k + n)^2), {n, 1, Infinity},
{k, 1, n}] == 3/8
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Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "1"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List["KroneckerDelta", "[", RowBox[List["1", ",", " ", RowBox[List["GCD", "[", RowBox[List["k", ",", "n"]], "]"]]]], "]"]], RowBox[List["k", " ", "n", SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "2"]]]]]]]], "\[Equal]", FractionBox["3", "8"]]]]]
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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> <munderover> <mo> ∑ </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mfrac> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mi> gcd </mi> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ] </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </msub> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mn> 3 </mn> <mn> 8 </mn> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mfrac> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mi> gcd </mi> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ] </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </msub> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mn> 3 </mn> <mn> 8 </mn> </mfrac> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n_", "=", "1"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "n_"], FractionBox[RowBox[List["KroneckerDelta", "[", RowBox[List["1", ",", RowBox[List["GCD", "[", RowBox[List["k_", ",", "n_"]], "]"]]]], "]"]], RowBox[List["k_", " ", "n_", " ", SuperscriptBox[RowBox[List["(", RowBox[List["k_", "+", "n_"]], ")"]], "2"]]]]]]]], "]"]], "\[RuleDelayed]", FractionBox["3", "8"]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
