The least prime number p such that p+2 is prime and p+6 has n prime factors (counted multiply).
5, 3, 239, 1619, 3119, 48119, 265619, 4379369, 20796869, 46484369, 517578119, 669921869
1
Note that 3 is the second term because 3+6 is 3*3.
T. D. Noe, Plot of 12 terms
D.R. Heath-Brown and Xiannan Li, Almost prime tripes and Chen's Theorem, arXiv 1504.0533 (Apr 02 2015).
(Mma) pc[n_] := Plus @@ Transpose[FactorInteger[n]][[2]]; nn = 10; t = Table[0, {nn}]; done = 0; n = 0; While[done < nn, n++; p1 = Prime[n]; p2 = Prime[n + 1]; If[p2 - p1 == 2, m = pc[p1 + 6]; If[m <= nn && t[[m]] == 0, t[[m]] = p1; done++]]]; t
Cf. S000555, S000556, S000563.
nonn
T. D. Noe, Apr 08 2015