Certain eta-quotients and ℓ-regular overpartitions

Let A ¯ ℓ ( n ) be the number of overpartitions of n into parts not divisible by ℓ . In this paper, we prove that A ¯ ℓ ( n ) is almost always divisible by p i j if p i 2 a i ≥ ℓ , where j is a fixed positive integer and ℓ = p 1 a 1 p 2 a 2 ⋯ p m a m with primes p i > 3 . We obtain a Ramanujan-type congruence for A ¯ 7 modulo 7. We also exhibit infinite families of congruences and multiplicative identities for A ¯ 5 ( n ) .