Arithmetic properties of MacMahon-type sums of divisors

TL;DR

This work analyzes arithmetic properties of MacMahon-type sums of divisors encoded by the generating function $U_t(a,q)$ for $a\in\{0,\pm1,\pm2\}$ and proves new Ramanujan-like congruences for the coefficients $MO(a,t;n)$. By exploiting a product decomposition $U_t(a,q)=\left(\sum b_k(a) q^k\right)\left(\sum c_n(a,t) q^{\binom{n+1}{2}}\right)$ with explicit $b_k(a)$ and $c_n(a,t)$, the authors derive numerous modular congruences modulo 3, 5, 7, 11, and higher powers, including results for overpartition-related counts $\overline{B}_3(n)$ and the negative-repth partition functions $p_{-3}(n)$. They extend known results and establish new infinite families, some of which arise from Lucas-type arguments and known congruences for partition-type functions, and they provide an elementary appendix proving $\overline{B}_3(15n+7)\equiv0\pmod{5}$. Overall, the paper advances the modular arithmetic of MacMahon-type divisor sums and expands the toolkit for Ramanujan-type congruences in partition theory.

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 $n$.

Theorems & Definitions (36)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof