New Ramanujan-type congruences for overpartitions modulo $11$ and $13$

Abstract

In this paper, we establish two new Ramanujan-type congruences for the overpartition function: $\overline{p}(11\times(8n+5))\equiv 0 \pmod{11}$ and $\overline{p}(13\times 2^6(8n+7))\equiv 0 \pmod{13}$. The proofs rely on the theory of modular forms. We conjecture potential Ramanujan-type congruences for overpartitions modulo 7, 17, 19 and 23.