Question 166
For an integer $n \geqslant 3$ we define the sequence $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{k}$ as the sequence of exponents in the prime factor decomposition of $n! = p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \ldots p_{k}^{\alpha_{k}}$, where $p_{1} \lt p_{2} \lt \cdots \lt p_{k}$ are primes.
Determine all integers $n \geqslant 3$ for which $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{k}$ is a geometric progression.
Determine all integers $n \geqslant 3$ for which $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{k}$ is a geometric progression.