 TOPICS # Digit Count

The number 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]. The number of 1s in the binary representation of a number , illustrated above, is given by   (1)   (2)

where 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   (3)   (4)

with , and by (5)

where is the denominator of (6)

For , 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).

For a binary number, the count of 1s is equal to the digit sum . The quantity is called the parity of a nonnegative integer . and satisfy the beautiful identities   (7)   (8)

where is the Euler-Mascheroni constant and (OEIS A094640) is its "alternating analog" (Sondow 2005).

Let and be the numbers of even and odd digits respectively of . Then   (9)   (10)   (11)

where the latter (OEIS A096614) is transcendental (Borwein et al. 2004, pp. 14-15).

Binary, Digit, Digit Product, Digit Sum, Parity, Stolarsky-Harborth Constant

## Related Wolfram sites

http://functions.wolfram.com/NumberTheoryFunctions/DigitCount/

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## References

Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.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. Reading, MA: Addison-Wesley, pp. 111-115, 1994.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 ." 1 Aug 2005. http://arxiv.org/abs/math.NT/0508042.Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, p. 33, 2004. http://www.mathematicaguidebooks.org/.Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 67, 1991.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 902, 2002.

Digit Count

## Cite this as:

Weisstein, Eric W. "Digit Count." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DigitCount.html