Arithmetic properties of MacMahon-type sums of divisors

TL;DR

This paper establishes new Ramanujan-like congruences for coefficients of an extended sum-of-divisors generating function and explores divisibility properties of almost 3-regular overpartitions.

Contribution

It introduces several new infinite families of congruences for MacMahon-type divisor sums and connects these to overpartition divisibility results.

Findings

Proved new Ramanujan-like congruences for $U_t(a,q)$ coefficients.

Established divisibility $ar{B}_3(15n+7) mod 5$ for all $n \\geq 0$.

Extended classical divisor sum functions with novel congruence properties.

Abstract

In this paper, we prove several new infinite families of Ramanujan--like congruences satisfied by the coefficients of the generating function $U_t(a,q)$ which is an extension of MacMahon's generalized sum-of-divisors function. As a by-product, we also show that, for all $n\geq 0$, $\overline{B}_3(15n+7)\equiv 0 \pmod{5}$ where $\overline{B}_3(n)$ is the number of almost $3$-regular overpartitions of $n$.