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arXiv · 2503.18574

A conjecture of Nadji, Ahmia and Ram\'{\i}rez on congruences for biregular overpartitions

Dazhao Tang
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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 $k\geq1$ and $n\geq1$, \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\'{\i}rez (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 $\overline{B}_{4,3}(n)$.

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Dazhao Tang. 2025-03-24. A conjecture of Nadji, Ahmia and Ram\'{\i}rez on congruences for biregular overpartitions. https://arxiv.org/abs/2503.18574

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