What interesting/useful infinite families of prime numbers are there? Right now it would be useful if I could find one with arbitrarily many 1's in its binary representation, but I am doing a larger problem with prime numbers where this might be needed.
One list of families of prime numbers is
which I have collected over several years. Most of these families are infinite though knowledge of their size is varied.
Perhaps worth noting is that, for any $n$, there are infinitely many primes with more than $n$ 1s in binary, since otherwise there could be only at most $k\choose n$ $k$-bit primes, in contradiction to the Prime Number Theorem.