Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











EulerPhi






Mathematica Notation

Traditional Notation









Number Theory Functions > EulerPhi[n] > Representations through equivalent functions > With related functions





http://functions.wolfram.com/13.06.27.0001.01









  


  










Input Form





EulerPhi[n] == Sum[1 - Sign[Sum[KroneckerDelta[Subscript[p, i], Subscript[q, j]], {i, 1, m}, {j, 1, s}]], {k, 1, n}] /; FactorInteger[n] == {{Subscript[p, 1], Subscript[n, 1]}, \[Ellipsis], {Subscript[p, m], Subscript[n, m]}} && FactorInteger[k] == {{Subscript[q, 1], Subscript[n, 1]}, \[Ellipsis], {Subscript[q, s], Subscript[n, s]}} && Element[n, Integers] && n > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EulerPhi", "[", "n", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List["(", RowBox[List["1", "-", RowBox[List["Sign", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "m"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "s"], RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["p", "i"], ",", SubscriptBox["q", "j"]]], "]"]]]]]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["FactorInteger", "[", "n", "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["p", "1"], ",", SubscriptBox["n", "1"]]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["p", "m"], ",", SubscriptBox["n", "m"]]], "}"]]]], "}"]]]], "\[And]", RowBox[List[RowBox[List["FactorInteger", "[", "k", "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["q", "1"], ",", SubscriptBox["n", "1"]]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["q", "s"], ",", SubscriptBox["n", "s"]]], "}"]]]], "}"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, EulerPhi] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> s </mi> </munderover> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <msub> <mi> p </mi> <mi> i </mi> </msub> <mo> , </mo> <msub> <mi> q </mi> <mi> j </mi> </msub> </mrow> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> factors </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> p </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> p </mi> <mi> m </mi> </msub> <mo> , </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> factors </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> q </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> q </mi> <mi> s </mi> </msub> <mo> , </mo> <msub> <mi> n </mi> <mi> s </mi> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> EulerPhi </ci> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Sign </ci> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> s </ci> </uplimit> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <ci> KroneckerDelta </ci> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> i </ci> </apply> <apply> <ci> Subscript </ci> <ci> q </ci> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <times /> <ci> factors </ci> <ci> n </ci> </apply> <list> <list> <apply> <ci> Subscript </ci> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </list> <ci> &#8230; </ci> <list> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> m </ci> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> m </ci> </apply> </list> </list> </apply> <apply> <eq /> <apply> <ci> factors </ci> <ci> k </ci> </apply> <list> <list> <apply> <ci> Subscript </ci> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </list> <ci> &#8230; </ci> <list> <apply> <ci> Subscript </ci> <ci> q </ci> <ci> s </ci> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> s </ci> </apply> </list> </list> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EulerPhi", "[", "n_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List["(", RowBox[List["1", "-", RowBox[List["Sign", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "m"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "s"], RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["p", "i"], ",", SubscriptBox["q", "j"]]], "]"]]]]]], "]"]]]], ")"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["FactorInteger", "[", "n", "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["p", "1"], ",", SubscriptBox["n", "1"]]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["p", "m"], ",", SubscriptBox["n", "m"]]], "}"]]]], "}"]]]], "&&", RowBox[List[RowBox[List["FactorInteger", "[", "k", "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["q", "1"], ",", SubscriptBox["n", "1"]]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["q", "s"], ",", SubscriptBox["n", "s"]]], "}"]]]], "}"]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2001-10-29





© 1998-2014 Wolfram Research, Inc.