Odd Divisor Function

The odd divisor function

(1)

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)
			(3)
			(4)

where  is defined to be 0 if  is odd. The generating function is given by

			(5)
			(6)
			(7)

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.

REFERENCES:

Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, p. 306, 2005.

Hirzebruch, F. Manifolds and Modular Forms, 2nd ed. Braunschweig, Germany: Vieweg, p. 133, 1994.

Riordan, J. Combinatorial Identities. New York: Wiley, p. 187, 1979.

Sloane, N. J. A. Sequences A000593/M3197, A001227, A050999, A051000, A051001, and A051002 in "The On-Line Encyclopedia of Integer Sequences."

Verhoeff, T. "Rectangular and Trapezoidal Arrangements." J. Integer Sequences 2, #99.1.6, 1999.a