Certain eta-quotients and arithmetic density of Andrews' singular overpartitions

In order to give overpartition analogues of Rogers-Ramanujan type theorems for the ordinary partition function, Andrews defined the so-called singular overpartitions. Singular over partition function (C) over bark,i(n) counts the number of overpartitions of n in which no part is divisible by k and only parts equivalent to +/- i (mod k) may be overlined. Andrews also proved two beautiful Ramanujan type congruences modulo 3 satisfied by (C) over bar (3,1)(n). Later on, Aricheta proved that for an infinite family of $, (C) over bar (3(n)) is almost always divisible by 2. In this article, for an infinite subfamily of $ considered by Aricheta, we prove that (C) over bar;(n) is almost always divisible by arbitrary powers of 2. We also prove that (C) over bar (()n) is almost always divisible by arbitrary powers of 3 when l = 3, 6, 12, 24. Proofs of our density results rely on the modularity of certain eta-quotients which arise naturally as generating functions for the Andrews' singular overpartition functions. (C) 2020 Elsevier Inc. All rights reserved.