2. For integers $n \geq 3$, define the sequence $\alpha_{1}, \alpha_{2}, \mathrm{~L}, \alpha_{k}$ as the exponents corresponding to the primes $p_{1}<p_{2}<\mathrm{L}<p_{k}$ in the prime factorization of $n!$, $n!=p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \mathrm{~L} p_{k}^{\alpha_{k}}$. Find all integers $n \geq 3$ such that $\alpha_{1}, \alpha_{2}, \mathrm{~L}, \alpha_{k}$ form a geometric sequence.