Arithmetic Properties modulo powers of $2$ and $3$ for Overpartition $k$-Tuples with Odd Parts

TL;DR

This paper investigates divisibility and congruence properties of overpartition k-tuples with odd parts, extending previous results with new congruences modulo powers of 3 and addressing a related conjecture.

Contribution

It introduces new congruences modulo powers of 3 for overpartition k-tuples with odd parts, expanding the understanding of their divisibility properties.

Findings

Proved several congruences modulo multiples of 3.

Established an infinite family of congruences modulo powers of 3.

Confirmed some cases of a conjecture by Saikia, Sarma, and Sellers.

Abstract

Recently, Drema and N. Saikia (2023) and M. P. Saikia, Sarma, and Sellers (2023) proved several congruences modulo powers of $2$ for overpartition triples with odd parts. In this paper, we study further divisibility properties of overpartition $k$-tuples with odd parts using elementary means as well as properties of modular forms. In particular, we prove several congruences modulo multiples of $3$, and an infinite family of congruences modulo powers of $3$; we also prove some cases of a conjecture of Saikia, Sarma, and Sellers.