Let $n > 1$ be a positive integer. Find all $n$-tuples $(a_1 , a_2 ,\ldots, a_n )$ of positive integers which are pairwise distinct, pairwise coprime, and such that for each $i$ in the range $1 \le i \le n$, \[(a_1 + a_2 + \ldots + a_n )|(a_1^i + a_2^i + \ldots + a_n^i )\].