Question 4
For a given integer $a_0 \gt 1$ one defines a sequence $(a_n)_{n \ge 0}$ as follows: for each $k \ge 0$, define $a_{k+1}$ as the smallest integer greater than $a_k$ such that $\text{gcd}(a_{k+1}, a_0 a_1 \cdots a_k) = 1$. Determine all values $a_0$ for which all terms $a_k$ are primes or powers of primes.