The 2-Adic Behavior of the Number of Partitions into Distinct Parts

Let Q(n) denote the number of partitions of an integer n into distinct parts. For positive integers j, the first author and B. Gordon proved that Q(n) is a multiple of 2j for every non-negative integer n outside a set with density zero. Here we show that if i 6≡ 0 (mod 2j), then #{0 ≤ n ≤ X : Q(n) ≡ i (mod 2)} j √ X / logX. In particular, Q(n) lies in every residue class modulo 2j infinitely often. In addition, we examine the behavior of Q(n) (mod 8) in detail, and we obtain a simple “closed formula” using the arithmetic of the ring Z[ √ −6].