Divisors: Primary definition (formula 13.02.02.0002)

Divisors

$Mathematica Notation$

Number Theory Functions

Divisors[n]

Primary definition

http://functions.wolfram.com/13.02.02.0002.01

Input Form

Divisors[n] == {1, Subscript[d, 2], \[Ellipsis], Subscript[d, m - 1], Subscript[d, m]} /; Element[Re[n], Integers] && Element[Im[n], Integers] && Element[Re[n/Subscript[d, k]], Integers] && Element[Im[n/Subscript[d, k]], Integers] && Element[Re[Subscript[d, k]], Integers] && Element[Im[Subscript[d, k]], Integers] && Re[Subscript[d, k]] <= Re[Subscript[d, k + 1]] && (Im[Subscript[d, k]] <= Im[Subscript[d, k + 1]] /; Re[Subscript[d, k]] == Re[Subscript[d, k + 1]]) && Subscript[d, 1] == 1 && 1 <= k <= m - 1 && m == DivisorSigma[0, n]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Divisors", "[", "n", "]"]], "\[Equal]", RowBox[List["{", RowBox[List["1", ",", SubscriptBox["d", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["d", RowBox[List["m", "-", "1"]]], ",", SubscriptBox["d", "m"]]], "}"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "n", "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Im", "[", "n", "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Re", "[", FractionBox["n", SubscriptBox["d", "k"]], "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Im", "[", FractionBox["n", SubscriptBox["d", "k"]], "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Re", "[", SubscriptBox["d", "k"], "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Im", "[", SubscriptBox["d", "k"], "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Re", "[", SubscriptBox["d", "k"], "]"]], "\[LessEqual]", RowBox[List["Re", "[", SubscriptBox["d", RowBox[List["k", "+", "1"]]], "]"]]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Im", "[", SubscriptBox["d", "k"], "]"]], "\[LessEqual]", RowBox[List["Im", "[", SubscriptBox["d", RowBox[List["k", "+", "1"]]], "]"]]]], "/;", RowBox[List[RowBox[List["Re", "[", SubscriptBox["d", "k"], "]"]], "\[Equal]", RowBox[List["Re", "[", SubscriptBox["d", RowBox[List["k", "+", "1"]]], "]"]]]]]], ")"]], "\[And]", RowBox[List[SubscriptBox["d", "1"], "\[Equal]", "1"]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["m", "-", "1"]]]], "\[And]", RowBox[List["m", "\[Equal]", RowBox[List["DivisorSigma", "[", RowBox[List["0", ",", "n"]], "]"]]]]]]]]]]

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> <mi> divisors </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <msub> <mi> d </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> d </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> d </mi> <mi> m </mi> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> n </mi> <msub> <mi> d </mi> <mi> k </mi> </msub> </mfrac> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> n </mi> <msub> <mi> d </mi> <mi> k </mi> </msub> </mfrac> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <mo> ≤ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> d </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <mo> ≤ </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> d </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> d </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> d </mi> <mn> 1 </mn> </msub> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> k </mi> <mo> ≤ </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ⩵ </mo> <mrow> <msub> <semantics> <mi> σ </mi> <annotation encoding='Mathematica'> TagBox["\[Sigma]", DivisorSigma] </annotation> </semantics> <mn> 0 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <ci> divisors </ci> <ci> n </ci> </apply> <list> <cn type='integer'> 1 </cn> <apply> <ci> Subscript </ci> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> d </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> m </ci> </apply> </list> </apply> <apply> <and /> <apply> <in /> <apply> <real /> <ci> n </ci> </apply> <integers /> </apply> <apply> <in /> <apply> <imaginary /> <ci> n </ci> </apply> <integers /> </apply> <apply> <in /> <apply> <real /> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <in /> <apply> <imaginary /> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <in /> <apply> <real /> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> </apply> <integers /> </apply> <apply> <in /> <apply> <imaginary /> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> </apply> <integers /> </apply> <apply> <leq /> <apply> <real /> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> </apply> <apply> <real /> <apply> <ci> Subscript </ci> <ci> d </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Condition </ci> <apply> <leq /> <apply> <imaginary /> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> </apply> <apply> <imaginary /> <apply> <ci> Subscript </ci> <ci> d </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <real /> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> </apply> <apply> <real /> <apply> <ci> Subscript </ci> <ci> d </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> d </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> k </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <eq /> <ci> m </ci> <apply> <ci> DivisorSigma </ci> <cn type='integer'> 0 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Divisors", "[", "n_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["{", RowBox[List["1", ",", SubscriptBox["d", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["d", RowBox[List["m", "-", "1"]]], ",", SubscriptBox["d", "m"]]], "}"]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "n", "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Im", "[", "n", "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Re", "[", FractionBox["n", SubscriptBox["d", "k"]], "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Im", "[", FractionBox["n", SubscriptBox["d", "k"]], "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Re", "[", SubscriptBox["d", "k"], "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Im", "[", SubscriptBox["d", "k"], "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Re", "[", SubscriptBox["d", "k"], "]"]], "\[LessEqual]", RowBox[List["Re", "[", SubscriptBox["d", RowBox[List["k", "+", "1"]]], "]"]]]], "&&", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Im", "[", SubscriptBox["d", "k"], "]"]], "\[LessEqual]", RowBox[List["Im", "[", SubscriptBox["d", RowBox[List["k", "+", "1"]]], "]"]]]], "/;", RowBox[List[RowBox[List["Re", "[", SubscriptBox["d", "k"], "]"]], "\[Equal]", RowBox[List["Re", "[", SubscriptBox["d", RowBox[List["k", "+", "1"]]], "]"]]]]]], ")"]], "&&", RowBox[List[SubscriptBox["d", "1"], "\[Equal]", "1"]], "&&", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["m", "-", "1"]]]], "&&", RowBox[List["m", "\[Equal]", RowBox[List["DivisorSigma", "[", RowBox[List["0", ",", "n"]], "]"]]]]]]]]]]]]

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

2001-10-29