[uckdlvbf] Composites in arithmetic progression

We are given a sequence a*n + b whose members are all not prime for 1 <= n <= nmax, where a and b are relatively prime.  (Note that a=1 or b=1 are permitted because they cause GCD = 1, which is the definition of relatively prime.)

What (a, b, nmax) are notable / difficult to find?

For the case a=1 (and a=2), this is about prime gaps, well studied by others.

This is the dual of Primes in Arithmetic Progression, also well studied by others.