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DigitCount






Mathematica Notation

Traditional Notation









Number Theory Functions > DigitCount[n,b] > Summation > Finite summation





http://functions.wolfram.com/13.10.23.0001.01









  


  










Input Form





Sum[DigitCount[k, 2, 1], {k, 0, n - 1}] == (n/2) Log[2, n] + n F[Log[2, n]] /; (F[n] == F[n + 1] && F[n] == 2^(Floor[Log[2, n]] - 1) (2 f[n/2^Floor[Log[2, n]] - 1] + (n/2^Floor[Log[2, n]]) Log[2, n/2^Floor[Log[2, n]]] - 2 (n/2^Floor[Log[2, n]] - 1)) && (f[x] == Sum[g[2^k x]/2^k, {k, 0, Infinity}] /; g[x] == (1/2) Mod[x, 1] UnitStep[1/2 - Mod[x, 1]] + (1/2) (1 - Mod[x, 1]) UnitStep[Mod[x, 1] - 1/2])) || (F[x] == Sum[Subscript[c, k] E^(2 Pi I k x), {k, 0, Infinity}] /; Subscript[c, 0] == Log[2, Pi]/2 - 1/(2 Log[2]) - 1/4 && Subscript[c, k] == (-(Log[2]/(2 I k Pi Log[2] - 4 k^2 Pi^2))) Zeta[(2 I k Pi)/Log[2]])










Standard Form





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MathML Form







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<apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> f </ci> <ci> x </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <ci> g </ci> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> g </ci> <ci> x </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <rem /> <ci> $CellContext`x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> UnitStep </ci> <apply> <plus /> <apply> <rem /> <ci> $CellContext`x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <rem /> <ci> $CellContext`x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> UnitStep </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <rem /> <ci> $CellContext`x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> EllipticF </ci> <ci> x </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> k </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <ci> k </ci> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <log /> <logbase> <cn type='integer'> 2 </cn> </logbase> <pi /> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> k </ci> <pi /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> k </ci> <pi /> <apply> <power /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </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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