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).