The least prime number p such that p+2 has up to two prime factors and p+6 has n prime factors.

3, 29, 379, 4999, 124349, 1616609, 59814749, 1854257399

1

Prime factors are not counted multiply. Hence, 3 is the first term because 3+6 has one prime factor, 3. And 1854257399 is the 8-th term because 1854257399+6 has 8 factors: 5*7*11*13*17*19*31*37.

T. D. Noe, Plot of 8 terms

D.R. Heath-Brown and Xiannan Li, Almost prime tripes and Chen's Theorem, arXiv 1504.0533 (Apr 02 2015).

(Mma) pc2[n_] := Length[FactorInteger[n]]; nn = 8; t = Table[0, {nn}]; done = 0; n = 1; While[done < nn, n++; p = Prime[n]; If[pc2[p + 2] <= 2, m = pc2[p + 6]; If[m <= nn && t[[m]] == 0, t[[m]] = p; done++]]]; t

Cf. S000557, S000558, S000562.

nonn

T. D. Noe, Apr 08 2015