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http://functions.wolfram.com/13.06.23.0001.01
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Sum[EulerPhi[k], {l, 1, n}, {k, 1, Floor[n/l]}] == (1/2) n (n + 1)
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Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l", "=", "1"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["Floor", "[", FractionBox["n", "l"], "]"]]], RowBox[List["EulerPhi", "[", "k", "]"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]]]]]]
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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> l </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mi> n </mi> <mi> l </mi> </mfrac> <mo> ⌋ </mo> </mrow> </munderover> <mrow> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", EulerPhi] </annotation> </semantics> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mi> n </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <ci> n </ci> <apply> <power /> <ci> l </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <sum /> <bvar> <ci> l </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> EulerPhi </ci> <ci> k </ci> </apply> </apply> </apply> <apply> <times /> <apply> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l_", "=", "1"]], "n_"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["Floor", "[", FractionBox["n_", "l_"], "]"]]], RowBox[List["EulerPhi", "[", "k", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", "n", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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