A343974 Even numbers k such that the two sets of primes in the Goldbach representation of k and k+2 as the sum of two odd primes do not intersect.

38, 68, 80, 98, 122, 128, 146, 158, 164, 188, 206, 212, 218, 224, 248, 278, 290, 302, 308, 326, 332, 338, 344, 368, 374, 380, 398, 410, 416, 428, 440, 458, 476, 488, 500, 518, 530, 536, 542, 548, 554, 578, 584, 608, 614, 626, 632, 638, 668, 674, 692, 698, 710

Offset: 1

Author

Giorgos Kalogeropoulos, Jun 07 2021

Keywords

• nonn

Comments

k is in the sequence iff the Goldbach representation of k as the sum of two odd primes does not contain any prime that is the lesser of a twin prime (A001359).

Conjecture: a(n) is congruent to 2 mod 6 with a(n)-3 not prime.

Examples

			The Goldbach representations of 80 and 82 as the sum of two odd primes are:
{{73, 7}, {67, 13}, {61, 19}, {43, 37}} and {{79, 3}, {71, 11}, {59, 23}, {53, 29}, {41, 41}}. The two sets {7, 13, 19, 37, 43, 61, 67, 73} and {3, 11, 23, 29, 41, 53, 59, 71, 79} do not intersect, so 80 is a term of the sequence.
		

Links

• Mahdi Meisami and Carlos Rivera, Puzzle 1040. Pair of consecutive even integers such that ..., The Prime Puzzles & Problems Connection.

Crossrefs

Cf. A001359, A046953.

Programs

Mathematica

Select[Range[6,1000,2],!IntersectingQ@@(Flatten@Select[IntegerPartitions[#,2],And@@PrimeQ[#]&]&/@{#,#+2})&]

Formula

a(n) = 6*A046953(n) + 2 (conjectured). - Hugo Pfoertner, Jun 09 2021