Charles Greathouse is right (sorry) -- P2s are more common than primes by a doubly-logarithmic factor. D. A. Goldston, S.W. Graham, J. Pintz, C.Y. Yilidirm: Small gaps between primes or almost primes http://arXiv.org/abs/math.NT/0506067 proves some results on small gaps between primes and between {primes U P2's}. It is trying (but failing) to prove an infinity of twin primes exists. Chen had proved there are infinitely many primes p with p+2 being either a prime or product of two primes. I however am not interested in shrinking the smallest gaps, but rather the largest gaps. The above paper does not discuss the latter.