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EulerPhi






Mathematica Notation

Traditional Notation









Number Theory Functions > EulerPhi[n] > Identities > Functional identities





http://functions.wolfram.com/13.06.17.0001.01









  


  










Input Form





EulerPhi[n] == (n/c[n]) EulerPhi[c[n]] /; c[n] == DivisorSum[Abs[MoebiusMu[d]] EulerPhi[d], {d, n}] == Product[Prime[k], {k, 1, r}] && n == Product[Subscript[p, k]^Subscript[n, k], {k, 1, r}] && Element[Subscript[p, k], Primes] && Element[Subscript[n, k], Integers] && Subscript[n, k] > 0










Standard Form





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MathML Form







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</mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <semantics> <mi> &#956; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Mu]&quot;, MoebiusMu] </annotation> </semantics> <mo> ( </mo> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, EulerPhi] </annotation> </semantics> <mo> ( </mo> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </munderover> <msubsup> <mi> p </mi> <mi> k </mi> <msub> <mi> n </mi> <mi> k </mi> </msub> </msubsup> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> p </mi> <mi> k </mi> </msub> <mo> &#8712; 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</ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> c </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> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> | </ms> <ms> n </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#62979; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#956; </ms> <ci> MoebiusMu </ci> </apply> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> ) </ms> </list> </apply> <ms> &#62980; </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#981; </ms> <ci> EulerPhi </ci> </apply> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> r </ms> </apply> <apply> <ci> SubsuperscriptBox </ci> <ms> p </ms> <ms> k </ms> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> k </ms> </apply> </apply> </list> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> p </ms> <ms> k </ms> </apply> <ms> &#8712; </ms> <apply> <ci> TagBox </ci> <ms> &#8473; </ms> <apply> <ci> Function </ci> <primes /> </apply> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> k </ms> </apply> <ms> &#8712; </ms> <apply> <ci> SuperscriptBox </ci> <ms> &#8469; </ms> <ms> + </ms> </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["EulerPhi", "[", "n_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["n", " ", RowBox[List["EulerPhi", "[", RowBox[List["c", "[", "n", "]"]], "]"]]]], RowBox[List["c", "[", "n", "]"]]], "/;", RowBox[List[RowBox[List[RowBox[List["c", "[", "n", "]"]], "\[Equal]", RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["MoebiusMu", "[", "d", "]"]], "]"]], " ", RowBox[List["EulerPhi", "[", "d", "]"]]]], ",", RowBox[List["{", RowBox[List["d", ",", "n"]], "}"]]]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "r"], RowBox[List["Prime", "[", "k", "]"]]]]]], "&&", RowBox[List["n", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "r"], SubsuperscriptBox["p", "k", SubscriptBox["n", "k"]]]]]], "&&", RowBox[List[SubscriptBox["p", "k"], "\[Element]", "Primes"]], "&&", RowBox[List[SubscriptBox["n", "k"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["n", "k"], ">", "0"]]]]]]]]]]










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





2001-10-29





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