β Scribed by Ken Ono; David Penniston
No coin nor oath required. For personal study only.
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 2 j for every non-negative integer n outside a set with density zero. Here we show that if i 0 (mod 2 j ), then
In particular, Q(n) lies in every residue class modulo 2 j 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].
π SIMILAR VOLUMES
In this paper, we find simple 2-adic congruences mod 2 [nΓ2]+1 for the No rlund numbers B (n) n and for the Bernoulli numbers of the second kind b n . These congruences improve F. T. Howard's mod 8 congruences (in ``Applications of Fibonacci Numbers, '' Vol. 5, pp. 355 366, Kluwer Academic, Dordrech
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