Arithmetic properties of MacMahon-type sums of divisors

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 $n\ge 0$, ${\bar{B}}_3(15n+7)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}5)$ where ${\bar{B}}_3(n)$ is the number of almost $3$-regular overpartitions of $n$.