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

AMS 11 · solved (literature)

theorem erdos_392 (A : ℕ → ℕ) (h : ∀ n > 0,
    IsLeast { t + 1 | (t) (_ : ∃ a : Fin (t + 1) → ℕ, (n)! = ∏ i, a i ∧
      Monotone a ∧ a (Fin.last t) ≤ n ^ 2) } (A n)) :
    ((fun (n : ℕ) => (A n - n / 2 + n / (2 * Real.log n) : ℝ)) =o[atTop] fun n => n / Real.log n)