|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/04.13.32.0001.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
q == DivisorProduct[(d + 1) CharacteristicFunction[d + 1, Primes],
{d, 2 n}] /; BernoulliB[2 n] == p/q && Element[n, Integers] && n >= 1
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["q", "\[Equal]", RowBox[List["DivisorProduct", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["d", "+", "1"]], ")"]], StyleBox[RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["d", "+", "1"]], ",", "Primes"]], "]"]], Rule[FontFamily, "Courier"]]]], ",", RowBox[List["{", RowBox[List["d", ",", RowBox[List["2", "n"]]]], "}"]]]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["BernoulliB", "[", RowBox[List["2", "n"]], "]"]], "\[Equal]", FractionBox["p", "q"]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "1"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> q </mi> <mo> ⩵ </mo> <mrow> <munder> <mo> ∏ </mo> <mrow> <mi> d </mi> <mo> | </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mrow> </munder> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> χ </mi> <mi> ℙ </mi> </msub> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation> </semantics> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msub> <mo> ⩵ </mo> <mrow> <mfrac> <mi> p </mi> <mi> q </mi> </mfrac> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mrow> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ≥ </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> ∏ </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> d </ms> <ms> | </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> d </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> χ </ms> <ms> ℙ </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> d </ms> <ms> + </ms> <ms> 1 </ms> </list> </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> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> B </ms> <ci> BernoulliB </ci> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> p </ms> <ms> q </ms> </apply> <ms> n </ms> </list> </apply> </list> </apply> <ms> ∈ </ms> <apply> <ci> SuperscriptBox </ci> <ms> ℕ </ms> <ms> + </ms> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> ≥ </ms> <ms> 1 </ms> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", "q_", "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["DivisorProduct", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["d", "+", "1"]], ")"]], " ", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List["d", "+", "1"]], ",", "Primes"]], "]"]]]], ",", RowBox[List["{", RowBox[List["d", ",", RowBox[List["2", " ", "n"]]]], "}"]]]], "]"]], "/;", RowBox[List[RowBox[List[RowBox[List["BernoulliB", "[", RowBox[List["2", " ", "n"]], "]"]], "\[Equal]", FractionBox["p", "q"]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "1"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|