A conjecture of Nadji, Ahmia and Ram\'ırez on congruences for biregular overpartitions

Abstract

Let ${\overline{B}}_{s,t}(n)$ denote the number of overpartitions of $n$ where no part is divisible by $s$ or $t$, with $s$ and $t$ being coprime. By establishing the exact generating functions of a family of arithmetic progressions in ${\overline{B}}_{4,3}(n)$, we prove that for any $k\ge 1$ and $n\ge 1$, \begin{align*} \overline{B}_{4,3}\big(2^{k+3}n\big)\equiv0\pmod{2^{3k+5}}. \end{align*} This significantly generalizes a conjectural congruence family posed by Nadji, Ahmia and Ram\'ırez (Ramanujan J. 67 (1):13, 2025) recently. Moreover, we conjecture that there is an infinite family of linear congruence relations modulo high powers of $2$ satisfied by ${\overline{B}}_{4,3}(n)$.

The main result of this paper (i.e., Theorem 1.1) has been proved by Adiga and Ranganatha (Discrete Math. (2018) 341, 3141--3147) in another equivalent form. Therefore, I think this manuscript should be withdrawn