On a problem of Chen and Liu concerning the prime power factorization of $n!$

Abstract:

For a fixed prime $p$, let $e_p(n!)$ denote the order of $p$ in the prime factorization of $n!$. Chen and Liu (2007) asked whether for any fixed $m$, one has $\{e_p(n^2!) \bmod m:\; n\in \mathbb {Z}\}=\mathbb {Z}_m$ and $\{e_p(q!) \bmod m:\; q \mbox { prime}\}=\mathbb {Z}_m$. We answer these two questions and show asymptotic formulas for $\# \{n<x: n \equiv a \bmod d,\; e_p(n^2!)\equiv r \bmod m\}$ and $\# \{q<x: q \mbox { prime}, q \equiv a \bmod d,\; e_p(q!)\equiv r \bmod m\}$. Furthermore, we show that for each $h\geqslant 3$, we have $\#\{n<x: n \equiv a \bmod d,\; e_p(n^h!)\equiv r \bmod m\} \gg x^{4/(3h+1)}$.