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 Ω( p + 2) ⩽ 3 and Ω( p + 6) ⩽ 6, where Ω( n ) denotes the total number of prime divisors of an integer n . We also prove a better conditional result, with the above Ω( p + 6) ⩽ 6 replaced by Ω( p + 6) ⩽ 3, under the Elliott-Halberstam conjecture.