  Recently, MacMahon's generalized sum-of-divisor functions were shown to link
partitions, quasimodular forms, and q-multiple zeta values. In this paper, we
explore many further properties and extensions of these. Firstly, we address a
question of Ono by producing infinite families of MacMahon-like functions that
approximate the colored partition functions (and indeed other eta quotients).
We further explore the MacMahon-like functions and discover new and suggestive
arithmetic structure and modular completions.
