Arithmetic properties of overpartitions into odd parts

In this article, we consider various arithmetic properties of the function (p(o)) over bar (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 (p(o)) over bar (n) and some easily-stated characterizations of (p(o)) over bar (n) modulo small powers of two. For example, it is proven that, for n >= 1, (p(o)) over bar (n) = 0 (mod 4) if and only if n is neither a square nor twice a square.