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%I #14 Jul 09 2021 21:46:56
%S 0,0,0,0,1,1,6,14,28,53,122,275,597,1203,2456,5111,10573,21662,44553,
%T 91246,185422,377264,765956,1552001,3140326,6349270,12825847,25891832
%N a(n) is the number of consecutive even prime gap pairs (g1, g2) satisfying g1 == 0 (mod 6) and g2 == 0 (mod 6) out of the first 2^n consecutive even prime gap pairs.
%C It seems that the fraction of prime gap pairs (g1, g2) for which g1 == 0 (mod 6), satisfying g2 == 0 (mod 6) as well, i.e., a(n)/A340948(n), tends to a constant, say c, when the number of prime gap pairs tends to infinity. From n = 27 we obtain that c > 0.431, while it can be argued heuristically that c < 0.5.
%C Meanwhile, the fractions of prime gap pairs (g1, g2), satisfying either g2 == 2 (mod 6) or g2 == 4 (mod 6), seem to tend both to another constant, (1-c)/2, when the number of prime gap pairs tends to infinity (see A345333 and A345334).
%F a(n) = A340948(n) - (A345333(n) + A345334(n)).
%Y Cf. A340948, A341531, A341532, A345333, A345334.
%K nonn,more
%O 0,7
%A _A.H.M. Smeets_, Jun 14 2021