A352240 Even numbers with at least one pair of Goldbach partitions, (p,q) and (r,s) with p,q,r,s prime and p < r <= s < q, such that all integers in the open intervals (p,r) and (s,q) are composite.

10, 16, 18, 22, 24, 30, 34, 36, 42, 46, 48, 54, 60, 64, 66, 72, 76, 78, 82, 84, 90, 96, 98, 102, 106, 108, 110, 112, 114, 120, 126, 132, 136, 138, 140, 142, 144, 150, 154, 156, 160, 162, 168, 174, 180, 184, 186, 188, 190, 192, 194, 196, 198, 202, 204, 210, 216, 218, 220, 222

Offset: 1

Author

Wesley Ivan Hurt, Mar 08 2022

Keywords

• nonn

Comments

Similar to A187797 but also contains the numbers 82, 96, 98, 110, 136, ...

Examples

			82 is in the sequence since it has at least one pair of Goldbach partitions, namely (23,59) and (29,53), such that all integers in the open intervals (23,29) and (53,59) are composite.
		

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. A187797, A278700, A352248, A352283.

Cf. A352442, A352443, A352444, A352445.

Programs

Mathematica

Table[If[Sum[Sum[KroneckerDelta[NextPrime[k], i]*KroneckerDelta[NextPrime[2 n - i], 2 n - k]*(PrimePi[k] - PrimePi[k - 1]) (PrimePi[2 n - k] - PrimePi[2 n - k - 1]) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {k, i}], {i, n}] > 0, 2 n, {}], {n, 150}] // Flatten

Formula

a(n) = A352442(n) + A352443(n).

a(n) = A352444(n) + A352445(n).