The partition function 
counts the partitions of
a nonnegative integer 
in which no parts are multiples
of the positive integer 
(Gordon and Ono 1997). It also counts the partitions
of 
into at most 
copies of each part, with 
.
There is a special case
 |
(1)
|
where 
is the partition function Q, and 
is the number of 
-modular irreducible
representations of the symmetric group 
when 
is a prime number. The generating function for 
is given by
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
is a q-Pochhammer symbol and 
.
The following table gives the first few values of 
for 
, 2, ... and small 
.
 | OEIS |  |
| 2 | A000009 | 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, ... |
| 3 | A000726 | 1, 2, 2, 4, 5, 7, 9, 13, 16, 22, 27, 36, 44, 57, ... |
| 4 | A001935 | 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, 38, 50, 64, 82, ... |
| 5 | A035959 | 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, 44, 60, 76, 100, ... |
For every nonnegative integer 
, Gordon and Ono (1997) show that
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
For positive integers 
and 
, define 
as the number of positive
integers 
for which 
.
Let 
have prime factorization 
. Gordon and Ono (1997) proved
that if 
,
then
 |
(7)
|
for every positive integer 
.
