Odd values of Q(n) are 1, 1, 3, 5, 27, 89, 165, 585, ... (OEIS A051044), and occur with ever decreasing frequency as n becomes large (unlike P(n), for which the fraction of odd values remains roughly 50%). This follows from the pentagonal number theorem which gives G(x) | congruent | product_(n = 1)^∞(1 + x^n) (mod 2) | congruent | product_(n = 1)^∞(1 - x^n) (mod 2) | congruent | sum_(n = - ∞)^∞ x^((3n^2 + n)/2) (mod 2) (Gordon and Ono 1997), so Q(n) is odd iff n is of the form k(3k ± 1)/2, i.e., 1, 5, 12, 22, 35, ... or 2, 7, 15, 26, 40, ....

partition function P | partition function P congruences | partition function Q