Erdős problem 392

Let $A(n)$ denote the least value of $t$ such that $$n! = a_1 \cdots a_t$$ with $a_1 \leq \cdots \leq a_t\leq n^2$. Then $$A(n) = \frac{n}{2} - \frac{n}{2\log n} + o\left(\frac{n}{\log n}\right).$$