Erdős Problem #392 — Factorizations of factorials with bounded factors

For each positive integer $n$, let $A(n)$ be the least value of $t+1$ for which there exist natural numbers $a_0,\ldots,a_t$ satisfying $a_0\le\cdots\le a_t\le n^2$ and

$$n!=\prod_{i=0}^{t}a_i.$$

Prove that

$$A(n)=\frac n2-\frac{n}{2\log n}+o\!\left(\frac n{\log n}\right) \qquad(n\to\infty).$$