Congruences for Andrews' (k, i)-singular overpartitions

Andrews recently defined new combinatorial objects which he called (k, i)-singular overpartitions and proved that they are enumerated by (C) over bar (k, i)(n) which is the number of overpartitions of n in which no part is divisible by k and only the parts equivalent to +/- i (mod k) may be overlined. Andrews further showed that (C) over bar (3,1)(n) satisfies some Ramanujan-type congruences modulo 3. In this paper, we show that for any pair (k, i), (C) over bar (k, i)(n) satisfies infinitely many Ramanujan-type congruences modulo any power of prime coprime to 6k. We also show that for an infinite family of k, the value (C) over bar (3k, k)(n) is almost always even. Finally, we investigate the parity of (C) over bar (4k, k).