For any $k\geq 2$ let $g_k(n)$ denote the maximum value of

$$(a_1+\cdots+a_k)-n$$

where $a_1,\ldots,a_k$ are integers such that $a_1!\cdots a_k! \mid n!$. Can one show that

$$\sum_{n\leq x}g_k(n) \sim c_k x\log x$$

for some constant $c_k$? Is it true that there is a constant $c_k$ such that for almost all $n<x$ we have

$$g_k(n)=c_k\log x+o(\log x)?$$