LAUSR

Let $${\overline{A}}_{\ell }(n)$$ be the number of overpartitions of n into parts not divisible by $$\ell $$ . In this paper, we prove that $${\overline{A}}_{\ell }(n)$$ is almost always divisible by $$p_i^j$$ if $$p_i^{2a_i}\ge \ell $$ , where j is a fixed positive integer and $$\ell =p_1^{a_1}p_2^{a_2} \dots p_m^{a_m}$$ with primes $$p_i>3$$ . We obtain a Ramanujan-type congruence for $${\overline{A}}_{7}$$ modulo 7. We also exhibit infinite families of congruences and multiplicative identities for $${\overline{A}}_{5}(n)$$ .