Number of times that 2*n can be written as the sum of a prime and a member of P2.

0, 0, 1, 1, 2, 1, 2, 2, 2, 3, 4, 3, 4, 3, 3, 4, 5, 4, 4, 4, 4, 5, 6, 5, 6, 6, 6, 6, 6, 6, 7, 6, 7, 6, 7, 8, 8, 9, 9, 8, 10, 9, 9, 8, 9, 9, 9, 9, 7, 9, 10, 10, 10, 11, 11, 12, 13, 12, 13, 13, 13, 10, 12, 10, 11, 13, 11, 11, 13, 13, 13, 14, 12, 12, 13, 12, 12, 15

1

P2 is the union of the primes and the numbers that are the product of two primes.

T. D. Noe, Plot of 5000 terms

T. D. Noe, Table of 5000 terms

Eric W. Weisstein, MathWorld: Chen’s Theorem

Tomohiro Yamada, Explicit Chen's theorem, arXiv 1511.03409 (Nov 11 2015)

(Mma) mx = 100; p = Prime[Range[2, 1 + PrimePi[mx]]]; p2 = Union[p, Select[Union[Flatten[Table[i*j, {i, p}, {j, p}]]], # <= mx &]]; t = Table[0, {p[[-1]]/ 2}]; Do[k = i + j; If[i >= j && k < p[[-1]], t[[k/2]]++], {i, p}, {j, p2}]; t

Cf. A002375 (sums of 2 primes).

nonn,nice

T. D. Noe, Nov 11 2015