A notorious question with prime numbers is estimating the gaps between consecutive primes. That is, if $(p_n)_{n \geq 1}$ is the canonical enumeration of the primes, then set $g_n = p_{n+1} - p_n$. It is shown that $g_n > \frac{c \log(n) \log \log(n) \log \log \log \log(n)}{(\log \log \log(n))^2}$ infinitely often, but a precise estimate is not known.

My question is, is there a 'natural' superset of the primes that are of interest (say, the set of numbers that are either primes or product of two primes) such that the gap between consecutive members is well known or well estimated?