Eta-quotients and divisibility of certain partition functions by powers of primes

Andrews' $(k, i)$-singular overpartition function $\overline{C}_{k, i}(n)$ 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. In recent times, divisibility of $\overline{C}_{3\ell, \ell}(n)$, $\overline{C}_{4\ell, \ell}(n)$ and $\overline{C}_{6\ell, \ell}(n)$ by $2$ and $3$ are studied for certain values of $\ell$. In this article, we study divisibility of $\overline{C}_{3\ell, \ell}(n)$, $\overline{C}_{4\ell, \ell}(n)$ and $\overline{C}_{6\ell, \ell}(n)$ by primes $p\geq 5$. For all positive integer $\ell$ and prime divisors $p\geq 5$ of $\ell$, we prove that $\overline{C}_{3\ell, \ell}(n)$, $\overline{C}_{4\ell, \ell}(n)$ and $\overline{C}_{6\ell, \ell}(n)$ are almost always divisible by arbitrary powers of $p$. For $s\in \{3, 4, 6\}$, we next show that the set of those $n$ for which $\overline{C}_{s\cdot\ell, \ell}(n) \not\equiv 0\pmod{p_i^k}$ is infinite, where $k$ is a positive integer satisfying $p_i^{k-1}\geq \ell$. We further improve a result of Gordon and Ono on divisibility of $\ell$-regular partitions by powers of certain primes. We also improve a result of Ray and Chakraborty on divisibility of $\ell$-regular overpartitions by powers of certain primes.