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Divisors






Mathematica Notation

Traditional 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





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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> &#8290; </mo> <mrow> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <msub> <mi> d </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </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> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> n </mi> <msub> <mi> d </mi> <mi> k </mi> </msub> </mfrac> <mo> ) </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> n </mi> <msub> <mi> d </mi> <mi> k </mi> </msub> </mfrac> <mo> ) </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> d </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </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> &#8289; </mo> <mo> ( </mo> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </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> &#8743; </mo> <mrow> <msub> <mi> d </mi> <mn> 1 </mn> </msub> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <mi> k </mi> <mo> &#8804; </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#10869; </mo> <mrow> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, 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> &#8230; </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





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





2001-10-29





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