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

## TL;DR

This paper proves that the singular overpartition function introduced by Andrews is almost always divisible by arbitrarily high powers of 2 and 3, extending previous divisibility results.

## Contribution

It establishes that for any positive integer k, the function C_{3,1}(n) is almost always divisible by 2^k and 3^k, generalizing earlier divisibility findings.

## Key findings

- C_{3,1}(n) is almost always divisible by 2^k for any k.
- C_{3,1}(n) is almost always divisible by 3^k for any k.
- Extends previous divisibility results for singular overpartitions.

## 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 $3^k.$