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

1,1

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.

Links

Table of n, a(n) for n=1..53.

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

Formula

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

Example

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.

Mathematica

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

Crossrefs

Cf. A001359, A046953.

Sequence in context: A216140 A177044 A321997 * A335483 A071885 A032506

Adjacent sequences: A343971 A343972 A343973 * A343975 A343976 A343977

Keyword

nonn

Author

Giorgos Kalogeropoulos, Jun 07 2021

Status

approved