FactorInteger[n] == {{-1, 1}, {Subscript[p, 1], Subscript[n, 1]}, {Subscript[p, 2], Subscript[n, 2]}, \[Ellipsis], {Subscript[p, m], Subscript[n, m]}} /; n == -Product[Subscript[p, k]^Subscript[n, k], {k, 1, m}] && Element[Subscript[p, k], Primes] && Element[Subscript[n, k], Integers] && Subscript[n, k] > 0 && Subscript[p, k] < Subscript[p, k + 1] && 1 <= k <= m - 1 && m == DivisorSigma[0, n]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["FactorInteger", "[", "n", "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["p", "1"], ",", SubscriptBox["n", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["p", "2"], ",", SubscriptBox["n", "2"]]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["p", "m"], ",", SubscriptBox["n", "m"]]], "}"]]]], "}"]]]], "/;", RowBox[List[RowBox[List["n", "\[Equal]", RowBox[List["-", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "m"], SubsuperscriptBox["p", "k", SubscriptBox["n", "k"]]]]]]]], "\[And]", RowBox[List[SubscriptBox["p", "k"], "\[Element]", "Primes"]], "\[And]", RowBox[List[SubscriptBox["n", "k"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["n", "k"], ">", "0"]], "\[And]", RowBox[List[SubscriptBox["p", "k"], "<", SubscriptBox["p", RowBox[List["k", "+", "1"]]]]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["m", "-", "1"]]]], "\[And]", RowBox[List["m", "\[Equal]", RowBox[List["DivisorSigma", "[", RowBox[List["0", ",", "n"]], "]"]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <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> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <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> <mrow> <mo> { </mo> <mrow> <msub> <mi> p </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> n </mi> <mn> 2 </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> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <msubsup> <mi> p </mi> <mi> k </mi> <msub> <mi> n </mi> <mi> k </mi> </msub> </msubsup> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> p </mi> <mi> k </mi> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8473; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalP]&quot;, Function[Primes]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> n </mi> <mi> k </mi> </msub> <mo> &#8712; </mo> <msup> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> p </mi> <mi> k </mi> </msub> <mo> &lt; </mo> <msub> <mi> p </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </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> factors </ci> <ci> n </ci> </apply> <list> <list> <cn type='integer'> -1 </cn> <cn type='integer'> 1 </cn> </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> <list> <apply> <ci> Subscript </ci> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </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> <and /> <apply> <eq /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <power /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> <primes /> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> <apply> <lt /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> <apply> <ci> Subscript </ci> <ci> p </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </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>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FactorInteger", "[", "n_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["p", "1"], ",", SubscriptBox["nn", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["p", "2"], ",", SubscriptBox["nn", "2"]]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["p", "m"], ",", SubscriptBox["nn", "m"]]], "}"]]]], "}"]], "/;", RowBox[List[RowBox[List["n", "\[Equal]", RowBox[List["-", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "m"], 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"]], "&&", RowBox[List[SubscriptBox["p", "k"], "<", SubscriptBox["p", RowBox[List["k", "+", "1"]]]]], "&&", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["m", "-", "1"]]]], "&&", RowBox[List["m", "\[Equal]", RowBox[List["DivisorSigma", "[", RowBox[List["0", ",", "n"]], "]"]]]]]]]]]]]]

2001-10-29