Due to the work of Yitang Zhang, James Maynard, Terence Tao and the Polymath8 project we know the current bound on prime gaps is 246. i.e, there are infinitely many pairs of primes that differ by a gap no more than $B, B \le 246$.
I was wondering if there is a systematic way to generate a subset of these primes. (We don't have to generate all, but the ones we generate should be pairs of primes with gap $\le 246$).
We could do sieving for primes in a range and then check for a prime $p$ in the range whether $p+246$ is also prime. Is there any other way?
EDIT: My understanding of the gap theorem was clarified in the comments section and I have edited the question to reflect that.