Famously, there are arbitrarily long arithmetic progressions $x$, $x+y$, ..., $x+ky$ consisting of primes, by Green-Tao. I was wondering whether the following generalization is also known (by the same methods or some extension):
Given positive integers $a_0,...,a_k$, does there exist an arithmetic progression $x$, $x+y$,..., $x+ky$ of integers such that $x+jy$ has exactly $a_j$ prime factors for each $j=0,...,k$? (Maybe the integers in the progression should also be demanded to be pairwise coprime, or otherwise cases like $a_0=...=a_k$ would be trivial by multiplying some constant into a Green-Tao progression.)
It seems at least logical, since the set of integers with exactly $n$ prime factors, $n\ge 2$, is somewhat ``more dense" than the set of prime numbers. On the other hand the fact that the members of the arithmetic progression no longer come from the same set might mess up the Szemeredi-type approach in Green-Tao's proof.
