%I #13 Jan 07 2019 19:22:14

%S 0,0,1,2,3,3,4,4,6,5,6,7,8,8,9,7,8,9,11,9,11,11,11,12,13,12,13,14,13,

%T 13,16,15,16,16,14,16,18,16,19,19,17,18,21,17,19,22,19,19,24,19,21,23,

%U 20,21,26,22,23,28,23,24,29,23,24,29,21,25,29,24,25,29,27,25,33,26,27,32,27

%N a(n) = number of primes of the form p = 2n - q, where q is a prime or semiprime.

%C Related to Chen's theorem (Chen 1966, 1973) which states that every sufficiently large even number is the sum of a prime and another prime or semiprime. Yamada (2015) has proved that this holds for all even numbers larger than exp(exp(36)).

%C In terms of this sequence, Chen's theorem with Yamada's bound is equivalent to say that a(n) > 0 for all n > 1.7 * 10^1872344071119348 (exponent ~ 1.8*10^15).

%C A235645 lists the number of decompositions of 2n into a prime p and a prime or semiprime q; this is less than a(n) because p + q and q + p is the same decomposition (if q is a prime), but this sequence will count two distinct primes 2n - q and 2n - p (if q <> p).

%C Sequence A322006 lists the same for even and odd numbers n, not only for even numbers 2n.

%D Chen, J. R. (1966). "On the representation of a large even integer as the sum of a prime and the product of at most two primes". Kexue Tongbao. 11 (9): 385-386.

%D Chen, J. R. (1973). "On the representation of a larger even integer as the sum of a prime and the product of at most two primes". Sci. Sinica. 16: 157-176.

%H Y. C. Cai, <a href="http://doi.org/10.1007/s101140200168">Chen's Theorem with Small Primes</a>. Acta Mathematica Sinica 18, no. 3 (2002), pp. 597-604. doi:10.1007/s101140200168.

%H P. M. Ross, <a href="http://doi.org/10.1112/jlms/s2-10.4.500">On Chen's theorem that each large even number has the form (p1+p2) or (p1+p2p3)</a>, J. London Math. Soc. Series 2 vol. 10, no. 4 (1975), pp. 500-506. doi:10.1112/jlms/s2-10.4.500.

%H Tomohiro Yamada, <a href="http://arxiv.org/abs/1511.03409">Explicit Chen's theorem</a>, preprint arXiv:1511.03409 [math.NT] (2015).

%F a(n) = A322006(2n).

%e a(4) = 2 since for n = 4, 2n = 8 = 2 + 6 = 3 + 5 = 5 + 3, i.e., primes 2, 3 and 5 are of the form specified in the definition (since 6 = 2*3 is a semiprime and 5 and 3 are primes).

%o (PARI) A322007(n,s=0)=forprime(p=2,-2+n*=2,bigomega(n-p)<3&&s++);s}

%Y Cf. A322006, A235645, A045917, A130588, A241539.

%K nonn

%O 0,4

%A _M. F. Hasler_, Jan 06 2019