Congruences modulo powers of 2 for t-colored overpartitions

Abstract

Let $\overline{p}_{-t}(n)$ count the number of t-colored overpartition of n, with t= 5, 7, 11 and 13. We find several infinite families of congruences modulo 16 and 32 for $\overline{p}_{-t}(n)$. For example, For each $\alpha$, $\beta$ and $\gamma \ge 0$,

$$\begin{aligned} \overline{p}_{-11}\left(8\cdot 3^{4\alpha }\cdot 5^{2\beta +2}\cdot 7^{2\gamma }n+t_5\cdot 3^{4\alpha }\cdot 5^{2\beta +1}\cdot 7^{2\gamma }\right)\equiv 0 \pmod {32}, \end{aligned}$$

where $t_5\in \{7, 23, 31, 39\}.$