Density results for the parity of (4k, k)-singular overpartitions

The (k, i)-singular overpartitions, combinatorial objects introduced by Andrews in 2015, are known to satisfy Ramanujan-type congruences modulo any power of prime coprime to 6k. In this paper we consider the parity of the number (C) over bar (k,i)(n) of (k, i)-singular overpartitions of n. In particular, we give a sufficient condition on even values of kso that the values of (C) over bar (4k,k)(n) are almost always even. Furthermore, we show that for odd values of k <= 23, k not equal 19, certain subsequences of (C) over bar (4k,k)(n) are almost always even. (c) 2024 Elsevier Inc. All rights reserved.