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%I #23 Jun 24 2021 02:46:39
%S 38,68,80,98,122,128,146,158,164,188,206,212,218,224,248,278,290,302,
%T 308,326,332,338,344,368,374,380,398,410,416,428,440,458,476,488,500,
%U 518,530,536,542,548,554,578,584,608,614,626,632,638,668,674,692,698,710
%N 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.
%C 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).
%C Conjecture: a(n) is congruent to 2 mod 6 with a(n)-3 not prime.
%H Mahdi Meisami and Carlos Rivera, <a href="https://www.primepuzzles.net/puzzles/puzz_1040.htm">Puzzle 1040. Pair of consecutive even integers such that ...</a>, The Prime Puzzles & Problems Connection.
%F a(n) = 6*A046953(n) + 2 (conjectured). - _Hugo Pfoertner_, Jun 09 2021
%e The Goldbach representations of 80 and 82 as the sum of two odd primes are:
%e {{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.
%t Select[Range[6,1000,2],!IntersectingQ@@(Flatten@Select[IntegerPartitions[#,2],And@@PrimeQ[#]&]&/@{#,#+2})&]
%Y Cf. A001359, A046953.
%K nonn
%O 1,1
%A _Giorgos Kalogeropoulos_, Jun 07 2021