Some new congruences for generalized overcubic partition function

Amdeberhan et al. (2024) introduced the notion of a generalized overcubic partition function ${\bar{a}}_c(n)$ and proved an infinite family of congruences modulo a prime $p\ge 3$ and some Ramanujan type congruences. In this paper, we show that ${\bar{a}}_{2^{\lambda }m+t}(n)\equiv {\bar{a}}_t(n)(\mathrm{m}\mathrm{o}\mathrm{d}{2}^{\lambda +1})$, where $\lambda \ge 1,m\ge 0,$ and $t\ge 1$ are integers. We also prove some new congruences modulo $8$ and $16$ for ${\bar{a}}_{2m+1}(n),{\bar{a}}_{2m+2}(n),{\bar{a}}_{8m+3}(n)$, where $m$ is any non-negative integer.