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 $n\ge 0$. 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$^k$.