A Simple Sample of Series

I knew that the sum of the sequence of reciprocals of integers (1 + 1/2 + 1/3 + 1/4 +... + 1/n) diverges to infinity.

Today on www.xkcd.com it was shown that the series of the reciprocals of prime numbers (1/2 + 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + ... + 1/P_i) also diverges to infinity.

I got to wondering which subsets of integers and subsequent reciprocals would diverge to infinity. For example, reciprocals of n^2 (1, 4, 9, 16, 25, ... , n^2) is a series that converges, not diverges.

So how about the fibonacci sequence? Twin Primes? All integers with exactly 3 prime factors?

If you know the answer to any of these or have interesting subsets of integers whose series of reciprocals diverge, let me know. 20 cool points may hang in the balance.