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

@article{Ono2000The2B,
  title={The 2-Adic Behavior of the Number of Partitions into Distinct Parts},
  author={Ken Ono and David Penniston},
  journal={J. Comb. Theory, Ser. A},
  year={2000},
  volume={92},
  pages={138-157}
}
Abstract 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≢0 (mod 2j), then #{0⩽n⩽X : Q(n)≡i (mod 2 j )}⪢ j X /log X. 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… 

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