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http://functions.wolfram.com/04.08.11.0001.01
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Sum[GCD[k, n] x^k, {k, 1, Infinity}] ==
DivisorSum[EulerPhi[d] (x^d/(1 - x^d)), {d, n}]
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Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[RowBox[List["GCD", "[", RowBox[List["k", ",", "n"]], "]"]], SuperscriptBox["x", "k"]]]]], "\[Equal]", RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List[RowBox[List["EulerPhi", "[", "d", "]"]], FractionBox[SuperscriptBox["x", "d"], RowBox[List["1", "-", SuperscriptBox["x", "d"]]]]]], ",", RowBox[List["{", RowBox[List["d", ",", "n"]], "}"]]]], "]"]]]]]]
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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> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> x </mi> <mi> k </mi> </msup> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <munder> <mo> ∑ </mo> <mrow> <mi> d </mi> <mo> | </mo> <mi> n </mi> </mrow> </munder> <mrow> <mrow> <mi> ϕ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mfrac> <msup> <mi> x </mi> <mi> d </mi> </msup> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mi> d </mi> </msup> </mrow> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∑ </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> ∞ </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> gcd </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> , </ms> <ms> n </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <ms> x </ms> <ms> k </ms> </apply> </list> </apply> </list> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> ∑ </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> d </ms> <ms> | </ms> <ms> n </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ϕ </ms> <ms> ( </ms> <ms> d </ms> <ms> ) </ms> </list> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> SuperscriptBox </ci> <ms> x </ms> <ms> d </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> SuperscriptBox </ci> <ms> x </ms> <ms> d </ms> </apply> </list> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "\[Infinity]"], RowBox[List[RowBox[List["GCD", "[", RowBox[List["k_", ",", "n_"]], "]"]], " ", SuperscriptBox["x_", "k_"]]]]], "]"]], "\[RuleDelayed]", RowBox[List["DivisorSum", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["EulerPhi", "[", "d", "]"]], " ", SuperscriptBox["x", "d"]]], RowBox[List["1", "-", SuperscriptBox["x", "d"]]]], ",", RowBox[List["{", RowBox[List["d", ",", "n"]], "}"]]]], "]"]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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