|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/13.05.17.0006.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
DivisorSigma[1, n] == DivisorSum[DivisorSigma[0, n/Subscript[d, j]]
EulerPhi[Subscript[d, j]], {Subscript[d, j], n}] /;
Element[Subscript[d, j], Divisors[n]] && Element[n, Integers] && n > 0
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["DivisorSigma", "[", RowBox[List["1", ",", "n"]], "]"]], "\[Equal]", RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List[RowBox[List["DivisorSigma", "[", RowBox[List["0", ",", FractionBox["n", SubscriptBox["d", "j"]]]], "]"]], RowBox[List["EulerPhi", "[", SubscriptBox["d", "j"], "]"]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["d", "j"], ",", "n"]], "}"]]]], "]"]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["d", "j"], "\[Element]", RowBox[List["Divisors", "[", "n", "]"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <semantics> <mi> σ </mi> <annotation encoding='Mathematica'> TagBox["\[Sigma]", DivisorSigma] </annotation> </semantics> <mn> 1 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <munder> <mo> ∑ </mo> <mrow> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> | </mo> <mi> n </mi> </mrow> </munder> <mrow> <mrow> <msub> <mi> σ </mi> <mn> 0 </mn> </msub> <mo> ( </mo> <mfrac> <mi> n </mi> <msub> <mi> d </mi> <mi> j </mi> </msub> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", EulerPhi] </annotation> </semantics> <mo> ( </mo> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> ∈ </mo> <mrow> <mi> divisors </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> σ </ms> <ci> DivisorSigma </ci> </apply> <ms> 1 </ms> </apply> <ms> ( </ms> <ms> n </ms> <ms> ) </ms> </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> <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> <apply> <ci> SubscriptBox </ci> <ms> σ </ms> <ms> 0 </ms> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> n </ms> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> ϕ </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> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> ∈ </ms> <apply> <ci> RowBox </ci> <list> <ms> divisors </ms> <ms> ( </ms> <ms> n </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> ∈ </ms> <apply> <ci> SuperscriptBox </ci> <ms> ℕ </ms> <ms> + </ms> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DivisorSigma", "[", RowBox[List["1", ",", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List[RowBox[List["DivisorSigma", "[", RowBox[List["0", ",", FractionBox["n", SubscriptBox["d", "j"]]]], "]"]], " ", RowBox[List["EulerPhi", "[", SubscriptBox["d", "j"], "]"]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["d", "j"], ",", "n"]], "}"]]]], "]"]], "/;", RowBox[List[RowBox[List[SubscriptBox["d", "j"], "\[Element]", RowBox[List["Divisors", "[", "n", "]"]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|