Divisibility of Andrews' Singular Overpartitions by Powers of 2 and 3

Abstract

Andrews introduced the partition function ${\overline{C}}_{k,i}(n)$, called singular overpartition, which counts the number of overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv \pm i(\mathrm{mod}k)$ may be overlined. He also proved that ${\overline{C}}_{3,1}(9n+3)$ and ${\overline{C}}_{3,1}(9n+6)$ are divisible by $3$ for $n\ge 0$. Recently Aricheta proved that for an infinite family of $k$, ${\overline{C}}_{3k,k}(n)$ is almost always even. In this paper, we prove that for any positive integer $k$, ${\overline{C}}_{3,1}(n)$ is almost always divisible by $2^k$ and $3^k.$

Accepted for publication at Research in Number Theory