A362640 Product of the larger primes, q, in the Goldbach partitions of 2n such that p + q = 2n, p <= q, and p,q prime (or 1 if no Goldbach partition of 2n exists).

1, 2, 3, 5, 35, 7, 77, 143, 143, 221, 3553, 4199, 5681, 391, 7429, 551, 351509, 392863, 589, 24679, 765049, 47027, 1175921, 58642669, 2318087, 55883, 95041567, 84323, 2961799, 5037203051, 78647, 367569469, 14263488419, 2257, 403723843, 22531226387, 461671607, 761740327

Offset: 1

Author

Wesley Ivan Hurt, Apr 28 2023

Keywords

• nonn
• easy

Examples

			a(10) = 221; 2*10 = 20 has two Goldbach partitions, namely 17+3 and 13+7. The product of the larger parts of these partitions, is 17*13 = 221.
		

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, A337568 (product of all prime parts), A362641 (product of smaller primes p).

Programs

Mathematica

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

Formula

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

a(n) = Product_{p+q = 2n, p<=q, and p,q prime} q.

a(n) = A337568(n) / A362641(n).