MEMO 2017 Problem 8

For an integer $n \geq 3$ we define the sequence $\alpha_1, \alpha_2, \ldots, \alpha_k$ as the sequence of exponents in the prime factorization of $n! = p_1^{\alpha_1}p_2^{\alpha_2} \ldots p_k^{\alpha_k}$ , where $p_1 < p_2 < \ldots < p_k$ are primes. Determine all integers $n \geq 3$ for which $\alpha_1, \alpha_2, \ldots, \alpha_k$ is a geometric progression.