Triples of almost primes

The k-tuple conjecture of Hardy and Littlewood predicts that there are infinitely many primes p such that p + 2 and p + 6 are primes simultaneously. In this paper, we prove that there are infinitely many primes p such that Omega(p + 2) <= 3 and Omega(p + 6) <= 6, where Omega(n) denotes the total number of prime divisors of an integer n. We also prove a better conditional result, with the above Omega(p + 6) <= 6 replaced by Omega(p + 6) <= 3, under the Elliott-Halberstam conjecture.