The odd divisor function 
is the sum of the 
th powers of
the positive odd divisors
of a positive integer 
, given by
 |
(1)
|
It is the analog of the divisor function for odd divisors only.
Subtracting the contribution of the even divisors gives
 |
(2)
|
where 
is the divisor function and 
is defined to be 0 if 
is odd. For the case 
,
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
The generating function is given by
 |  |  |
(6)
|
 |  | ![1/(24)[theta_3^4(x)+theta_2^4(x)-1]](/images/equations/OddDivisorFunction/Inline22.svg) |
(7)
|
 |  |  |
(8)
|
for 
,
where 
denotes a Jacobi theta function.
The value 
equals the number of distinct cyclotomic polynomials
in the polynomial factorization of 
over the rationals.
The following table gives the first few 
for 
, 2, ....
 | OEIS |  |
| 0 | A001227 | 1, 1, 2, 1, 2, 2, 2, 1, 3, 2, ... |
| 1 | A000593 | 1, 1, 4, 1, 6, 4, 8, 1, 13, 6, ... |
| 2 | A050999 | 1, 1, 10, 1, 26, 10, 50, 1, 91, 26, ... |
| 3 | A051000 | 1, 1, 28, 1, 126, 28, 344, 1, 757, 126, ... |
| 4 | A051001 | 1, 1, 82, 1, 626, 82, 2402, 1, 6643, 626, ... |
| 5 | A051002 | 1, 1, 244, 1, 3126, 244, 16808, 1, 59293, 3126, ... |
This function arises in Ramanujan's Eisenstein series 
and in a recurrence
relation for the partition function P.
