divisibility and k-power sum

Question

Let $a_{1},\dots,a_{n},\,n>2$ distinct natural numbers. Prove that if $p_{1},\dots,p_{r}$ are prime numbers and they divide $a_{1}+\dots+a_{n}$ then exists an integer $k>1$ and a prime $p\neq p_{i},\,\forall i=1,\dots,r$ such that $p\mid a_{1}^{k}+\dots+a_{n}^{k}.$ I think it can be seen like $a_{1}+\dots+a_{n}\nmid a_{1}^{k}+\dots+a_{n}^{k}\,\,\forall k\in\mathbb{N}$ but I don't know how attack the problem.