RMM 2013 Problem 5

Given an integer $k \geq 2$, set $a_{1}=1$ and, for every integer $n \geq 2$, let $a_{n}$ be the smallest $x>a_{n-1}$ such that:

$$x=1+\sum_{i=1}^{n-1}\left\lfloor\sqrt[k]{\frac{x}{a_{i}}}\right\rfloor .$$

Prove that every prime occurs in the sequence $a_{1}, a_{2}, \ldots$. (Bulgaria) Alexander Ivanov