Given some integer-valued irreducible polynomial $P$, define a sequence $S$ by including the smallest positive $n$ such that $P(n)$ is pairwise coprime to every element in the sequence.
For example, if $P(x):=x^2+2x+2$, we have
$$S = \{5, 17, 26, 37, 101, \ldots\}.$$
My impression after experimentation is that these sequences, assuming the polynomial has no common factors etc., contain terms having no more than $\deg(P)$ prime factors in every term beyond some bound $M$; that is, there are sometimes a couple of exceptions early on, but this seems to hold in the limit. If true, it seems to also invite a stronger version which says that terms with $\deg(P)$ factors will occur infinitely many times.
It's somewhat computation intensive to check, particularly with higher-degree polynomials, so I'm wondering if anyone knows whether this is already an established result.