Erdős Problem #390 — Let $f(n)$ be the minimal $m$ such that $n! = a_1\cdots a_k$ with $n< a_1<\cdots <a_k=m$.

Let $f(n)$ be the minimal $m$ such that $n! = a_1\cdots a_k$ with $n< a_1<\cdots <a_k=m$. Is there (and what is it) a constant $c$ such that $f(n)-2n \sim c\frac{n}{\log n}?$