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

Andrews’ $(k,i)$-singular overpartition function ${\bar{C}}_{k,i}(n)$ counts the number of overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv \pm i(\mathrm{m}\mathrm{o}\mathrm{d}k)$ may be overlined. In recent times, divisibility of ${\bar{C}}_{3\mathrm{\ell },\mathrm{\ell }}(n)$, $\overline{C}{4\ell, \ell}(n)$and$\overline{C}{6\ell, \ell}(n)$by$2$and$3$ are studied for certain values of $\mathrm{\ell }$. In this article, we study divisibility of ${\bar{C}}_{3\mathrm{\ell },\mathrm{\ell }}(n)$, ${\bar{C}}_{4\mathrm{\ell },\mathrm{\ell }}(n)$ and ${\bar{C}}_{6\mathrm{\ell },\mathrm{\ell }}(n)$ by primes $p\ge 5$. For all positive integer $\mathrm{\ell }$ and prime divisors $p\ge 5$ of $\mathrm{\ell }$, we prove that ${\bar{C}}_{3\mathrm{\ell },\mathrm{\ell }}(n)$, ${\bar{C}}_{4\mathrm{\ell },\mathrm{\ell }}(n)$ and ${\bar{C}}_{6\mathrm{\ell },\mathrm{\ell }}(n)$ 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 ${\bar{C}}_{s\cdot \mathrm{\ell },\mathrm{\ell }}(n)\equiv ̸0(\mathrm{m}\mathrm{o}\mathrm{d}{p}_i^k)$ is infinite, where $k$ is a positive integer satisfying $p_i^{k-1}\ge \mathrm{\ell }$. We further improve a result of Gordon and Ono on divisibility of $\mathrm{\ell }$-regular partitions by powers of certain primes. We also improve a result of Ray and Chakraborty on divisibility of $\mathrm{\ell }$-regular overpartitions by powers of certain primes.