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# 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.

## See also

Divisor Function, Even Divisor Function

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## 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.

## Referenced on Wolfram|Alpha

Odd Divisor Function

## Cite this as:

Weisstein, Eric W. "Odd Divisor Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OddDivisorFunction.html