TOPICS
Search

Odd Divisor Function

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

The odd divisor function

sigma_k^((o))(n)=sum_(d|n; d odd)d^k
(1)

is the sum of kth powers of the odd divisors of a number n. It is the analog of the divisor function for odd divisors only.

For the case k=1,

sigma_1^((o))(n)=sum_(d|n; d odd)d
(2)
=sum_(d|n)((-1)^(d+1)n)/d
(3)
=sigma_1(n)-2sigma_1(n/2),
(4)

where sigma_k(n/2) is defined to be 0 if n is odd. The generating function is given by

sum_(n=0)^(infty)sigma_1^((o))(n)x^n=sum_(n=0)^(infty)(nx^n)/(1+x^n)
(5)
=1/(24)[theta_3^4(x)+theta_2^4(x)]
(6)
=x+x^2+4x4+6x^5+4x^6+8x^7+...,
(7)

where theta_n(q) is a Jacobi elliptic function.

Rather surprisingly, sigma_0^((o))(n) gives the number of factors of the polynomial a^n+1.

The following table gives the first few sigma_k^((o))(n).

kOEISsigma_k^((o))(n)
0A0012271, 1, 2, 1, 2, 2, 2, 1, 3, 2, ...
1A0005931, 1, 4, 1, 6, 4, 8, 1, 13, 6, ...
2A0509991, 1, 10, 1, 26, 10, 50, 1, 91, 26, ...
3A0510001, 1, 28, 1, 126, 28, 344, 1, 757, 126, ...
4A0510011, 1, 82, 1, 626, 82, 2402, 1, 6643, 626, ...
5A0510021, 1, 244, 1, 3126, 244, 16808, 1, 59293, 3126, ...

This function arises in Ramanujan's Eisenstein series L(q) and in a recurrence relation for the partition function P.

See also

Divisor Function, Even Divisor Function

Explore with Wolfram|Alpha

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

Subject classifications