Prime series

Source: https://cms.math.ca/crux/v40/n1/public_synopsis_v40n1.pdf

A sequence {an : n ≥ 2} is called prime-picking if, for each n, an is a prime divisor of n. A sequence {an : n ≥ 2} is called spread-out if, for each positive integer k, there is an index N such that, for n ≥ N, the k consecutive entries

a _ n, a _ (n+1), · · · , a _ (n+k−1) are all distinct.

Does there exist a prime-picking spread-out sequence?