The number Wolfram Language as DigitCount n , b , d ].
The number of 1s binary representation of a number
where greatest dividing exponent of 2 with respect to power of a prime factorial (Vardi 1991, Graham et al. 1994). Writing recurrence relation
with
(5)
where denominator of
(6)
For A000120 ; Smith 1966, Graham 1970, McIlroy 1974).
For a binary number, the count of 1s digit sum parity of a nonnegative integer
where Euler-Mascheroni constant and A094640 ) is its "alternating analog" (Sondow 2005).
Let
where the latter (OEIS A096614 ) is transcendental (Borwein et al. 2004, pp. 14-15).
See also Binary ,
Digit ,
Digit Product ,
Digit Sum ,
Parity ,
Stolarsky-Harborth Constant Related Wolfram sites http://functions.wolfram.com/NumberTheoryFunctions/DigitCount/ Explore with Wolfram|Alpha References Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Graham, R. L. "On Primitive Graphs and Optimal Vertex Assignments." Ann. New York Acad. Sci. 175 , 170-186, 1970. Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Factorial Factors." §4.4 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. McIlroy, M. D. "The Number of 1's in Binary Integers: Bounds and Extremal Properties." SIAM J. Comput. 3 , 255-261, 1974. Sloane, N. J. A. Sequences A000120 /M0105, A094640 , A096614 in "The On-Line Encyclopedia of Integer Sequences." Smith, N. "Problem B-82." Fib. Quart. 4 , 374-365, 1966. Sondow, J. "New Vacca-type Rational Series for Euler's Constant and its 'alternating' Analog http://arxiv.org/abs/math.NT/0508042 . Trott, M. The Mathematica GuideBook for Programming. http://www.mathematicaguidebooks.org/ . Vardi, I. Computational Recreations in Mathematica. Wolfram, S. A New Kind of Science. 902 , 2002. Referenced on Wolfram|Alpha Digit Count Cite this as: Weisstein, Eric W. "Digit Count." From MathWorld https://mathworld.wolfram.com/DigitCount.html
Subject classifications More... Less... 
of digits 
in the base-
representation of a number 
is called the 
-ary digit count for 
. The digit count is implemented in the Wolfram Language as DigitCount[n, b, d]. 
in the binary representation of a number 
, illustrated above, is given by 
is the greatest dividing exponent of 2 with respect to 
. This is a special application of the general result that the power of a prime 
dividing a factorial (Vardi 1991, Graham et al. 1994). Writing 
for 
, the number of 1s is also given by the recurrence relation 
, and by 
is the denominator of ![1/(n!)[(d^n)/(dx^n)(1-x)^(-1/2)]_(x=0).](https://mathworld.wolfram.com/images/equations/DigitCount/NumberedEquation2.svg)
, 2, ..., the first few values are 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, ... (OEIS A000120; Smith 1966, Graham 1970, McIlroy 1974). 
is equal to the digit sum 
. The quantity 
is called the parity of a nonnegative integer 
. 
and 
satisfy the beautiful identities 
is the Euler-Mascheroni constant and 
(OEIS A094640) is its "alternating analog" (Sondow 2005). 
and 
be the numbers of even and odd digits respectively of 
. Then 
." 1 Aug 2005.