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

1,1

Comments

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

Links

Table of n, a(n) for n=1..60.

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

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

Example

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.

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

Crossrefs

Cf. A187797, A278700, A352248, A352283.

Cf. A352442, A352443, A352444, A352445.

Sequence in context: A104868 A103208 A323196 * A187797 A352297 A192221

Adjacent sequences: A352237 A352238 A352239 * A352241 A352242 A352243

Keyword

nonn

Author

Wesley Ivan Hurt, Mar 08 2022

Status

approved