Elementary Proofs of Recent Congruences for Overpartitions Wherein Non-Overlined Parts are Not Divisible by 6

Abstract

We define $\overline{R_l^{\ast }}(n)$ as the number of overpartitions of $n$ in which non-overlined parts are not divisible by $l$. In a recent work, Nath, Saikia, and the second author established several families of congruences for $\overline{R_l^{\ast }}(n)$, with particular focus on the cases $l=6$ and $l=8$. In the concluding remarks of their paper, they conjectured that $\overline{R_6^{\ast }}(n)$ satisfies an infinite family of congruences modulo $128$. In this paper, we confirm their conjectures using elementary methods. Additionally, we provide elementary proofs of two congruences for $\overline{R_6^{\ast }}(n)$ previously proven via the machinery of modular forms by Alanazi, Munagi, and Saikia.