USAMO 2024 Problem 1

Find all integers $n \geq 3$ such that the following property holds: if we list the divisors of $n!$ in increasing order as $1 = d_1 < d_2 < \dots < d_k = n!$ , then we have $$d_2 - d_1 \leq d_3 - d_2 \leq \dots \leq d_k - d_{k-1}.$$

Proposed by Luke Robitaille.