Problem 400

Erdős Problem #400: declared status 'open'. Formalized: yes. 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$$g_k(n)=c_k\log x+o(\log x)?$$ Current best: Erd\H{o}s 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. Prize: no. Tags: factorials, number theory.