All Random Solved Random Open
For any $k\geq 2$ let $g_k(n)$ denote the maximal value of \[n-(a_1+\cdots+a_k)\] 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 \[g_k(n)=c_k\log x+o(\log x)?\]
Erdős and Graham write that it is easy to show that $g_k(n) \ll_k \log n$ always, but the best possible constant is unknown.

See also [401].