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%I A346776 #22 Sep 16 2021 02:24:30
%S A346776 0,0,0,0,2,3,6,15,28,58,132,254,515,1042,2088,4172,8337,16720,33556,
%T A346776 66948,134088,268037,535435,1069932,2139357,4275948,8544351,17076036
%N A346776 a(n) is the number of consecutive even prime gaps (g1, g2) satisfying g1 == 2 (mod 6) and g2 == 0 (mod 6) out of the first 2^n consecutive even prime gaps.
%C A346776 The prime gaps are given in A001223. Here look at terms of A001223 satisfying the conditions A001223(k) == 2 and A001223(k+1) == 0 (mod 6) for 1 < k <= 2^n + 1.
%F A346776 a(n) = A341531(n) - A346777(n) - 1.
%F A346776 a(n) - A345334(n) is in {0, 1}. This holds not only for powers of 2 counts, but for all counts.
%e A346776 The sequence A001223(n) mod 6 is given by: 1, 2, 2, 4, 2, 4, 2, 4, 0, 2, 0, 4, 2, 4, 0, 0, 2, 0, 4, 2, 0, 4, 0, 2, ..., denoted here as b(0), b(1), b(2), ..., i.e. b(n) = A001223(n+1) (mod 6) for n >= 0.
%e A346776 The term b(0) is excluded by definition. The conditions b(k) = 2 and b(k+1) == 0 are obtained for k = 9, 16, 19, ...
%e A346776 So a(n) = 0 for n <= 3 (the first value of k is 9, i.e. larger than 2^3), and a(4) = 2 (two values of k satisfying k <= 2^4).
%Y A346776 Cf. A001223, A340948, A341531, A341532, A345332, A345333, A345334, A346777.
%K A346776 nonn,more
%O A346776 0,5
%A A346776 _A.H.M. Smeets_, Aug 03 2021

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