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http://functions.wolfram.com/13.10.03.0003.01
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DigitCount[n, 2, 1] ==
2 n - Log[2, Denominator[(1/n!) Derivative[n][(1 - #1)^(-(1/2)) & ][0]]]
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Cell[BoxData[RowBox[List[RowBox[List["DigitCount", "[", RowBox[List["n", ",", "2", ",", "1"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["2", "n"]], "-", RowBox[List["Log", "[", RowBox[List["2", ",", RowBox[List["Denominator", "[", RowBox[List[FractionBox["1", RowBox[List["n", "!"]]], RowBox[List[RowBox[List[RowBox[List["Derivative", "[", "n", "]"]], "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "#"]], ")"]], "^", RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]]]], "&"]], "]"]], "[", "0", "]"]]]], "]"]]]], "]"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mi> s </mi> <mn> 2 </mn> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> - </mo> <mrow> <msub> <mi> log </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mi> q </mi> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mfrac> <mi> p </mi> <mi> q </mi> </mfrac> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msup> <mo> ∂ </mo> <mi> n </mi> </msup> <mrow> <mo> ∂ </mo> <msup> <mi> x </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> x </mi> </mrow> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> </mi> </mrow> </mrow> <msub> <mo> | </mo> <mrow> <mi> x </mi> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> </msub> <mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> </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> SubsuperscriptBox </ci> <ms> s </ms> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <ms> 1 </ms> <ms> ) </ms> </list> </apply> </apply> <ms> ( </ms> <ms> n </ms> <ms> ) </ms> </list> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> log </ms> <ms> 2 </ms> </apply> <ms> q </ms> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> p </ms> <ms> q </ms> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> ! </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> SuperscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> ∂ </ms> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ∂ </ms> <apply> <ci> SuperscriptBox </ci> <ms> x </ms> <ms> n </ms> </apply> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> SqrtBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> x </ms> </list> </apply> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> </ms> </list> </apply> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> | </ms> <apply> <ci> RowBox </ci> <list> <ms> x </ms> <ms> ⩵ </ms> <ms> 0 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> gcd </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> p </ms> <ms> , </ms> <ms> q </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ⩵ </ms> <ms> 1 </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DigitCount", "[", RowBox[List["n_", ",", "2", ",", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["Log", "[", RowBox[List["2", ",", RowBox[List["Denominator", "[", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "#1"]], ")"]], RowBox[List["-", FractionBox["1", "2"]]]], "&"]], ")"]], TagBox[RowBox[List["(", "n", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "0", "]"]], RowBox[List["n", "!"]]], "]"]]]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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