  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 $3^k.$

Number Theory