A362641 Product of the smaller primes, p, 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, 3, 15, 5, 21, 15, 35, 21, 165, 385, 273, 55, 1001, 39, 2805, 7735, 133, 561, 13585, 273, 5865, 124355, 5187, 1265, 391391, 741, 27115, 19605131, 1767, 64515, 5766215, 217, 374187, 12212915, 313131, 170085, 142635185, 63973, 902451, 13147103255, 223041, 101065, 818183948197

Offset

1,2

Links

Table of n, a(n) for n=1..45.

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

Formula

a(n) = Product_{k=1..n} 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} p.

a(n) = A337568(n) / A362640(n).

Example

a(10) = 21; 2*10 = 20 has two Goldbach partitions, namely 17+3 and 13+7. The product of the smaller parts of these partitions, is 3*7 = 21.

Mathematica

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

Crossrefs

Cf. A010051, A045917, A337568 (product of all prime parts), A362640 (product of the larger primes q).

Sequence in context: A039793 A106243 A109203 * A039792 A076358 A208798

Adjacent sequences: A362638 A362639 A362640 * A362642 A362643 A362644

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 28 2023

Status

approved