Question 91
Given an integer $k \ge 2$, set $a_1 = 1$ and, for every integer $n \ge 2$, let $a_n$ be the smallest $x \gt a_{n-1}$ such that
$$
x = 1 + \sum_{i=1}^{n-1} \lfloor \sqrt[k]{\frac{x}{a_i}} \rfloor.
$$
Prove that every prime occurs in the sequence $a_1, a_2, \dots$.
$$
x = 1 + \sum_{i=1}^{n-1} \lfloor \sqrt[k]{\frac{x}{a_i}} \rfloor.
$$
Prove that every prime occurs in the sequence $a_1, a_2, \dots$.