The odd divisor function
(1)
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is the sum of th powers of the odd divisors of a number . It is the analog of the divisor function for odd divisors only.
For the case ,
(2)
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(3)
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(4)
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where is defined to be 0 if is odd. The generating function is given by
(5)
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(6)
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(7)
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where is a Jacobi elliptic function.
Rather surprisingly, gives the number of factors of the polynomial .
The following table gives the first few .
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.