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 $$A(n) = \frac{n}{2} - \frac{n}{2\log n} + o\left(\frac{n}{\log n}\right).$$

1∀ (A : ℕ → ℕ),2  (∀ n > 0,3      IsLeast {x | ∃ t, ∃ (_ : ∃ a, n.factorial = ∏ i, a i ∧ Monotone a ∧ a (Fin.last t) ≤ n ^ 2), t + 1 = x} (A n)) →4    (fun n => ↑(A n) - ↑n / 2 + ↑n / (2 * Real.log ↑n)) =o[Filter.atTop] fun n => ↑n / Real.log ↑n