Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.6

A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four

Michael D. Hirschhorn
School of Mathematics and Statistics
Sydney 2052

James A. Sellers
Department of Mathematics
Penn State University
University Park, PA 16802


In 2001, Andrews and Lewis utilized an identity of F. H. Jackson to derive some new partition generating functions as well as identities involving some of the corresponding partition functions. In particular, for 0 < a < b < k, they defined W1(a, b; k; n) to be the number of partitions of n in which the parts are congruent to a or b mod k and such that, for any j, kj + a and kj + b are not both parts. Our primary goal in this note is to prove that W1(1,3;4;27n + 17) ≡ 0 (mod 3) for all n ≥ 0. We prove this result using elementary generating function manipulations and classic results from the theory of partitions.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequences A000122 A000203 A001935 A010054 A070048 A121455.)

Received June 29 2014; revised versions received September 2 2014; September 4 2014. Published in Journal of Integer Sequences, September 4 2014.

Return to Journal of Integer Sequences home page