Congruences and asymptotics of Andrews' singular overpartitions

Recently, singular overpartitions   were defined and studied by G.E. Andrews. He showed that such partitions can be enumerated by C‾k,i(n), the number of overpartitions of n such that no part is divisible by k   and only parts ≡±i(modk) may be overlined. Andrews proved some congruences for C‾3,1(n)(mod3). The author, M.D. Hirschhorn and J.A. Sellers found infinite families of congruences for C‾3,1(n), C‾4,1(n), C‾6,1(n) and C‾6,2(n). Z. Ahmed and N.D. Baruah obtained some new congruences for C‾3,1(n), C‾8,2(n), C‾12,2(n), C‾12,4(n), C‾24,8(n) and C‾48,16(n). In this paper, we prove some new congruences for C‾3,1(n) and C‾4,1(n) modulo powers of 2 and congruences of C‾k,i(n) for a family of pairs k,ik,i. We also obtain an asymptotic formula for C‾k,i(n) as n tends to infinity.