OPEN

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
\[f(n)-2n \sim c\frac{n}{\log n}?\]

Erdős, Guy, and Selfridge

[EGS82] have shown that
\[f(n)-2n \asymp \frac{n}{\log n}.\]