On Some New Congruences For Biregular Overpartitions

TL;DR

This paper investigates the arithmetic properties of overpartition functions avoiding multiples of two integers, establishing new congruences modulo specific numbers for cases where these integers are powers of 2 and 3.

Contribution

It introduces new congruences for overpartition counts with parts not divisible by certain powers of 2 and 3, extending previous results and employing generating functions and theta functions.

Findings

Established congruences modulo 4, 8, 6, 12 for overpartition counts

Extended previous results to broader cases with powers of 2 and 3

Used generating functions, dissection formulas, and theta functions

Abstract

Inspired by the recent work by Nadji, Ahmia and Ram\'irez, we examined the arithmetic properties of $\bar{B}_{l_1,l_2} (n)$, the number of overpartitions of n whose parts are neither divisible by $l_1$ nor divisible by $l_2$. In particular, we establish some congruences modulo k in {4, 8, 6, 12} satisfied by $\bar{B}_{l_1,l_2} (n)$ where $l_1$ and $l_2$ take values as arbitrary powers of 2 and 3. Moreover, we extend certain results proved in [26] and [15] for $l_1$ and $l_2$ with random powers of 2 and 3. Generating functions, dissection formulas, and theta functions are used to prove our main findings.