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%I #15 Oct 07 2022 23:59:49
%S 5,89,809,3954889,15186319,77011289,288413159,62585146739,
%T 114058236679,143014298809,2320700383121,2942277160583,4386778243843,
%U 16104860854333,23171469790747
%N Primes that begin a record list of consecutive primes whose sum of any 2 consecutive terms is divisible by 3.
%C The Rivera link gives the first 15 terms.
%H Carlos Rivera, <a href="https://www.primepuzzles.net/puzzles/puzz_1059.htm">Puzzle 1059 Set of consecutive primes such that...</a>, The Prime Puzzles and Problems Connection.
%e A001043 begins as (5, 8, 12, 18, 24, 30, 36, 42, 52) where 7 consecutive terms are multiple of 3 starting at 3rd term, corresponding to 5=prime(3), so 5 is the 1st term of this sequence.
%t p=2;t=r=0;Monitor[Do[p=NextPrime[p];t=0;While[Mod[p+NextPrime@p,3]==0,p=NextPrime@p;t++];If[t>r,Print[NextPrime[p,-t]];r=t],{n,10^14}],p] (* _Giorgos Kalogeropoulos_, Oct 24 2021 *)
%Y Cf. A001043, A253969.
%K nonn,more
%O 1,1
%A _Michel Marcus_, Oct 24 2021