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 GCD

 http://functions.wolfram.com/04.08.11.0001.01

 Input Form

 Sum[GCD[k, n] x^k, {k, 1, Infinity}] == DivisorSum[EulerPhi[d] (x^d/(1 - x^d)), {d, n}]

 Standard Form

 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"]], "}"]]]], "]"]]]]]]

 MathML Form

 k = 1 gcd ( k , n ) x k d | n ϕ ( d ) x d 1 - x d FormBox RowBox RowBox UnderoverscriptBox RowBox k = 1 RowBox RowBox gcd ( RowBox k , n ) SuperscriptBox x k RowBox UnderscriptBox ErrorBox RowBox d | n RowBox RowBox ϕ ( d ) FractionBox SuperscriptBox x d RowBox 1 - SuperscriptBox x d TraditionalForm [/itex]

 Rule Form

 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"]], "}"]]]], "]"]]]]]]

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

 2001-10-29