Congruences modulo 64 and 1024 for overpartitions

Abstract

Recently, several infinite families of congruences modulo 32, 64 and 256 for $\overline{p}(n)$ have been established by Yang et al. where $\overline{p} (n)$ denotes the number of overpartitions of n. In this paper, we establish congruences modulo 64 and 1024 by using identities for $r_3(n)$ and $r_7(n)$, where $r_k(n)$ is the number of representations of n as a sum of k squares. For example, we prove that for $n,\ \alpha \ge 0$,

$$\begin{aligned} \overline{p}( 3^{16\alpha +15}(24n+5) ) \equiv \overline{p}( 3^{16\alpha +15}(24n+13) ) \equiv 0\ (\mathrm{mod}\ 1024). \end{aligned}$$

In particular, we generalize some congruences for $\overline{p}(n)$ due to Yang et al.