cp's OEIS Frontend

A322006 a(n) = number of primes of the form p = n - q, where q is a prime or semiprime.

%I A322006 #25 Jun 24 2025 00:57:40
%S A322006 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 A322006 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 A322006 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 A322006 a(n) = number of primes of the form p = n - q, where q is a prime or semiprime.
%C A322006 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 A322006 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 A322006 Sequence A322007(n) = a(2n) lists the bisection corresponding to even numbers only.
%C A322006 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 A322006 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 A322006 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 A322006 David A. Corneth, <a href="/A322006/b322006.txt">Table of n, a(n) for n = 0..10000</a>
%H A322006 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 A322006 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 A322006 Tomohiro Yamada, <a href="http://arxiv.org/abs/1511.03409">Explicit Chen's theorem</a>, preprint arXiv:1511.03409 [math.NT] (2015).
%H A322006 Veritasium, <a href="https://youtu.be/x32Zq-XvID4&amp;t=1958">The Simplest Unsolved Problem in Math</a>, YouTube video (2025).
%e A322006 a(4) = 1 is the first nonzero term corresponding to 4 = 2 + 2 or, rather, to the prime 2 = 4 - 2.
%e A322006 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 A322006 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 A322006 (PARI) A322006(n,s=0)={forprime(p=2,n-2,bigomega(n-p)<3&&s++);s}
%Y A322006 Cf. A322007, A235645, A045917, A130588, A241539.
%K A322006 nonn,nice
%O A322006 0,6
%A A322006 _M. F. Hasler_, Jan 06 2019