The digit sum 
of a nonnegative integer 
is the sum of its base-
digits, where the base 
is an integer.
The base-10 digit sum of the integer 
is implemented in the Wolfram
Language as DigitSum[n],
and the base-
digit sum as DigitSum[n, b].
The following table gives 
for 
, 2, ... and small 
.
 | OEIS |  for  , 2, ... |
| 2 | A000120 | 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, ... |
| 3 | A053735 | 1, 2, 1, 2, 3, 2, 3, 4, 1, 2, 3, 2, 3, 4, 3, ... |
| 4 | A053737 | 1, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6, ... |
| 5 | A053824 | 1, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, ... |
| 6 | A053827 | 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, ... |
| 7 | A053828 | 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 2, 3, ... |
| 8 | A053829 | 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, ... |
| 9 | A053830 | 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, ... |
| 10 | A007953 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, ... |
Plots of the digit sums of first thousand positive integers are illustrated above for bases 2 to 10.
Plotting 
versus 
and 
gives the plot shown above.
The digit sum 
satisfies the congruence
 |
(1)
|
In base 10, this congruence is the basis of casting out nines and of fast divisibility tests such as those for 3 and 9.
For integers 
and 
,
 |
(2)
|
Indeed, 
splits the product into two digit blocks of length
, including leading zeros, whose corresponding digits
sum to 
(Carfì 2026, pp. 19-20).
This identity underlies a construction of Harshad numbers
in base 
.
satisfies the following unexpected
identity
 |
(3)
|
the 
case of which was given in the 1981
Putnam competition (Allouche 1992). In addition,
 |  |  |
(4)
|
![sum_(n=2)^(infty)[s_2(n)]^2(8n^3+4n^2+n-1)/(4n(n^2-1)(4n^2-1))](/images/equations/DigitSum/Inline28.svg) |  |  |
(5)
|
(OEIS A100044 and A100045; Allouche 1992, Allouche and Shallit 1992).
Let 
be the number of digit
blocks of 11 in the binary expansion of 
, then
 |  |  |
(6)
|
(OEIS A100046; Allouche 1992).
Sondow (2006) noted the unexpected identity
 |
(7)
|
The special case of 
corresponds to a Thue-Morse sequence product
(J. Sondow, pers. comm., Oct. 31, 2006).
The numbers 1, 81, 1458 and 1729 (OEIS A110921) are each the product of their own digit sum and its reversal, for example 
,
and 
. These are the only four numbers with
this property, as proved by Fujiwara (Fujiwara and Ogawa 2005).
For a fixed integer base 
, an integer 
can be the digit sum of a square
number only if 
for some integer 
. Call such a 
admissible. For every sufficiently large admissible 
, vibefrtz (2026) proved that there are at least 
positive
integers 
satisfying 
, 
, and 
; the stated bound holds whenever 
.
The construction and the lower bound 
were formally verified in Lean.
