A278700 Number of Goldbach partitions (p,q) of 2n such that all primes from p to q (inclusive) appear as a part in some Goldbach partition of p+q = 2n.

0, 0, 1, 1, 2, 1, 1, 0, 2, 0, 1, 3, 1, 0, 3, 0, 1, 2, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 5, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 3, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 0, 1

Offset: 1

Author

Wesley Ivan Hurt, Nov 26 2016

Keywords

• nonn
• easy

Comments

Records are a(3) = 1, a(5) = 2, a(12) = 3, a(30) = 5, a(165) = 6, a(8021811) = 7. - Charles R Greathouse IV, Nov 30 2016

a(n) <= A002375(n). - Wesley Ivan Hurt, Dec 17 2016

Examples

			a(5) = 2; There are 2 Goldbach partitions of 2*5 = 10, namely (3,7) and (5,5). (3,7) satisfies the condition that every prime from 3 to 7 (inclusive) appears as a part in some Goldbach partition of 10. This is also true of (5,5) since 5 appears in its own partition.
		

Links

• Eric Weisstein's World of Mathematics, Goldbach Partition
• Wikipedia, Goldbach's conjecture
• Index entries for sequences related to Goldbach conjecture
• Index entries for sequences related to partitions

Crossrefs

Cf. A000720, A002375, A010051, A055381, A279103.

Programs

Mathematica

Table[Sum[Times @@ Map[Boole@ PrimeQ@ # &, {i, 2 n - i}] * Product[(1 - Abs[Subtract @@ Map[Boole@ PrimeQ@ # &, {k, 2 n - k}]]), {k, i, n}], {i, 3, n}], {n, 89}] (* Michael De Vlieger, Nov 30 2016 *)

PARI

a(n) = sum(i=3, n, (ispseudoprime(i) * ispseudoprime(2*n-i) * prod(k=i, n, (1-abs(ispseudoprime(k)-ispseudoprime(2*n-k)))))) \\ Felix Fröhlich, Nov 28 2016

PARI

a(n) = if(n<3, return(0)); my(s,p=n,N=2*n); forprime(q=n, N, while(pCharles R Greathouse IV, Nov 30 2016

Formula

a(n) = Sum_{i=3..n} (c(i) * c(2*n-i) * Product_{k=i..n} (1 - abs(c(k) - c(2*n-k)))), where c is the prime characteristic (A010051).