The 
-ary digit count 
is the number of occurrences of the digit 
in the base-
representation of a positive
integer 
, where 
and 
are integers. The digit
count is implemented in the Wolfram Language
as DigitCount[n,
b, d].
The number of 1s 
in the binary representation of a positive
integer 
, 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 greatest
dividing exponent of a prime 
with respect to 
is 
(Vardi 1991, Graham et al. 1994). Writing 
for 
, the number of 1s is also
given, for 
, by the recurrence
relation
 |  |  |
(3)
|
 |  |  |
(4)
|
with 
, and by
 |
(5)
|
where 
is the denominator of
![1/(n!)[(d^n)/(dx^n)(1-x)^(-1/2)]_(x=0).](/images/equations/DigitCount/NumberedEquation2.svg) |
(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 
.
Writing 
for the number of 0s in the binary representation of 
without leading zeros, 
and 
satisfy the 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 decimal digits,
respectively, of 
. Then
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
where the latter (OEIS A096614) is transcendental (Borwein et al. 2004, pp. 14-15).
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1 Aug 2005.