Number of abelian groups of order $n$ as a product of partition numbers

Abelian groups of order $\prod p_k^{e_k}$ are counted by the product of the partition numbers $P(e_k)$.Suggest a slogan

Let $n = \prod_{k=1}^m p_k^{e_k}$, where $p_1, \ldots, p_m$ are distinct primes and $e_1, \ldots, e_m \geq 1$, and for an integer $e \geq 1$ let $P(e)$ be the number of partitions of $e$. Up to isomorphism, there are exactly $\prod_{k=1}^m P(e_k)$ abelian groups of order $n$.