[math-fun] Generalization of modified twin primes conjecture

Last month, Zhang proved that there exists a number N such that there are infinitely many primes that differ from another prime by not more than N. (He showed that N is at most 70 million. That upper bound has since been reduced to 12,012. See http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_pri... ) I've wondered if the same is true for any monotonically increasing sequence of positive integers (i.e. no duplicate terms) for which the sum of the reciprocals diverges. Can anyone think of a counterexample?