Arithmetic Properties of Overpartitions into Odd Parts

Abstract

In this article, we consider various arithmetic properties of the function $\overline{{p_{o} }} (n)$ 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 $\overline{{p_{o} }} (n)$ and some easily-stated characterizations of $\overline{{p_{o} }} (n)$ modulo small powers of two. For example, it is proven that, for n ≥ 1, $\overline{{p_{o} }} (n) \equiv 0$ (mod 4) if and only if n is neither a square nor twice a square.