A337568 Product of all the parts in the Goldbach partitions (p,q) of 2n such that p + q = 2n, p <= q, and p,q prime (or 1 if no Goldbach partition of 2n exists).

1, 4, 9, 15, 525, 35, 1617, 2145, 5005, 4641, 586245, 1616615, 1550913, 21505, 7436429, 21489, 985982745, 3038795305, 78337, 13844919, 10393190665, 12838371, 6896776665, 7292509103495, 12023917269, 70691995, 37198413949697, 62483343, 80309179885, 98755025688454681, 138969249

Offset: 1

Author

Wesley Ivan Hurt, Sep 29 2020

Keywords

• nonn

Examples

			a(9) = 5005; 2*9 = 18 has Goldbach partitions (13,5) and (11,7). The product of all the parts is 13 * 5 * 11 * 7 = 5005.
		

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. A010051, A045917, A238711, A362640 (product of the larger primes q), A362641 (product of the smaller primes p).

Programs

Mathematica

Table[Product[(i*(2 n - i))^((PrimePi[i] - PrimePi[i - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1])), {i, n}], {n, 40}]

Formula

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

a(n) = A362640(n) * A362641(n).