CONO-SUR 2018 Problem 6

A sequence $a_1, a_2,\dots, a_n$ of positive integers is alagoana, if for every $n$ positive integer, one have these two conditions I- $a_{n!} = a_1\cdot a_2\cdot a_3\cdots a_n$ II- The number $a_n$ is the $n$ -power of a positive integer. Find all the sequence(s) alagoana.