Partition Function Q Congruences

Odd values of  are 1, 1, 3, 5, 27, 89, 165, 585, ... (OEIS A051044), and occur with ever decreasing frequency as  becomes large (unlike , for which the fraction of odd values remains roughly 50%). This follows from the pentagonal number theorem which gives

			(1)
			(2)
			(3)

(Gordon and Ono 1997), so  is odd iff  is of the form , i.e., 1, 5, 12, 22, 35, ... or 2, 7, 15, 26, 40, ....

The values of  for which  is prime are 3, 4, 5, 7, 22, 70, 100, 495, 1247, 2072, 320397, ... (OEIS A035359), with no others for  (Weisstein, May 6, 2000). These values correspond to 2, 2, 3, 5, 89, 29927, 444793, 602644050950309, ... (OEIS A051005). It is not known if  is infinitely often prime, but Gordon and Ono (1997) proved that it is "almost always" divisible by any given power of 2 (1997).

Gordon and Hughes (1981) showed that

(4)

and

(5)

where  is an integer depending only on .

REFERENCES:

Gordon, B. and Hughes, K. "Ramanujan Congruences for ." In Analytic Number Theory, Proceedings of the Conference Held at Temple University, Philadelphia, Pa., May 12-15, 1980 (Ed. M. I. Knopp). New York: Springer-Verlag, pp. 333-359, 1981.

Gordon, B. and Ono, K. "Divisibility of Certain Partition Functions by Powers of Primes." Ramanujan J. 1, 25-34, 1997.

Sloane, N. J. A. Sequences A035359, A051005, and A051044 in "The On-Line Encyclopedia of Integer Sequences."