Arithmetic Properties of Overpartitions into Odd Parts

Michael D. Hirschhorn^{1} and James A. Sellers^{2}

m.hirschhorn@unsw.edu.au

sellersj@math.psu.edu

Annals of Combinatorics 10 (3) p. 353-367 September, 2006

Abstract:

In this article, we consider various arithmetic properties of the function which
denotes the number of overpartitions of n using only odd parts. This function has arisen in a
number of recent papers, but in contexts which are very different from overpartitions. We prove
a number of arithmetic results including several Ramanujan-like congruences satisfied by
and some easily-stated characterizations of modulo small powers of two. For example, it is
proven that, for n ≥ 1, ≡ 0 (mod 4) if and only if n is neither a square nor twice a square.

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