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

%I #14 Jul 09 2019 05:54:53

%S 0,0,0,0,1,2,2,3,3,4,3,3,4,4,4,4,6,5,5,4,6,5,7,4,8,5,8,5,9,4,7,4,8,7,

%T 9,4,11,5,9,6,11,6,11,6,11,8,12,4,13,6,12,8,13,6,14,5,13,8,13,4,16,5,

%U 15,9,16,7,16,6,14,9,16,5,18,6,16,10,19,7,19,6,17,10,18,4,21,9,17,9,19,8

%N a(n) = number of primes of the form p = n - 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(2*n) > 0 for all n > 1.7 * 10^1872344071119348 (exponent ~ 1.8*10^15).

%C Sequence A322007(n) = a(2n) lists the bisection corresponding to even numbers only.

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

%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).

%e a(4) = 1 is the first nonzero term corresponding to 4 = 2 + 2 or, rather, to the prime 2 = 4 - 2.

%e a(5) = 2 because the primes 2 = 5 - 3 and 3 = 5 - 2 are of the required form n - q where q = 3 resp. q = 2 are primes.

%e a(6) = 2 because the primes 2 = 6 - 4 and 3 = 6 - 3 are of the required form n - q, since q = 4 is a semiprime and q = 3 is a prime.

%o (PARI) A322006(n,s=0)=forprime(p=2,n-2,bigomega(n-p)<3&&s++);s}

%Y Cf. A322007, A235645, A045917, A130588, A241539.

%K nonn

%O 0,6

%A _M. F. Hasler_, Jan 06 2019