A241539 -id:A241539 - cp's OEIS frontend

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

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, 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, 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

Offset: 0

M. F. Hasler, Jan 06 2019

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(2*n) > 0 for all n > 1.7 * 10^1872344071119348 (exponent ~ 1.8*10^15).
Sequence A322007(n) = a(2n) lists the bisection corresponding to even numbers only.
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).

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

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.

David A. Corneth, Table of n, a(n) for n = 0..10000
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).
Veritasium, The Simplest Unsolved Problem in Math, YouTube video (2025).

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

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

A322007 a(n) = number of primes of the form p = 2n - q, where q is a prime or semiprime.

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Jan 06 2019

nonn

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.

			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).

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.

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).

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

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

a(n) = A322006(2n).

cp's OEIS Frontend

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

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