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 $\equiv \pm i\pmod{k}$ 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 $n\geq 0$. 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 $3^k.$