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 A322007 a(n) = number of primes of the form p = 2n - q, where q is a prime or semiprime. 2
 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, 13, 16, 15, 16, 16, 14, 16, 18, 16, 19, 19, 17, 18, 21, 17, 19, 22, 19, 19, 24, 19, 21, 23, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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)). 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). 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). Sequence A322006 lists the same for even and odd numbers n, not only for even numbers 2n. REFERENCES 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. 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. LINKS Table of n, a(n) for n=0..76. Y. C. Cai, Chen's Theorem with Small Primes. Acta Mathematica Sinica 18, no. 3 (2002), pp. 597-604. doi:10.1007/s101140200168. P. M. Ross, On Chen's theorem that each large even number has the form (p1+p2) or (p1+p2p3), J. London Math. Soc. Series 2 vol. 10, no. 4 (1975), pp. 500-506. doi:10.1112/jlms/s2-10.4.500. Tomohiro Yamada, Explicit Chen's theorem, preprint arXiv:1511.03409 [math.NT] (2015). FORMULA a(n) = A322006(2n). EXAMPLE 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). PROG (PARI) A322007(n, s=0)=forprime(p=2, -2+n*=2, bigomega(n-p)<3&&s++); s} CROSSREFS Cf. A322006, A235645, A045917, A130588, A241539. Sequence in context: A127434 A266475 A205402 * A226107 A365659 A344870 Adjacent sequences: A322004 A322005 A322006 * A322008 A322009 A322010 KEYWORD nonn AUTHOR M. F. Hasler, Jan 06 2019 STATUS approved

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Last modified December 11 10:53 EST 2023. Contains 367722 sequences. (Running on oeis4.)