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