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%I #16 Sep 16 2021 02:31:23
%S 0,1,2,3,4,9,16,27,56,111,187,373,708,1403,2780,5467,10781,21248,
%T 41701,82581,163473,323995,643327,1278401,2540048,5050955,10052647,
%U 20010073
%N a(n) is the number of consecutive even prime gaps (g1, g2) satisfying g1 == 2 (mod 6) and g2 == 4 (mod 6) out of the first 2^n consecutive even prime gaps.
%C The prime gaps are given in A001223. Here we consider the gaps satisfying the conditions A001223(k) == 2 and A001223(k+1) == 4 (mod 6) for 1 < k <= 2^n + 1.
%F a(n) = A341531(n) - A346776(n) - 1.
%e The sequence A001223(n) mod 6 is given by:
%e 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 The term b(0) is excluded by definition. The conditions b(k) = 2 and b(k+1) == 4 are obtained for k = 2, 4, 6, 12 ...
%e So a(0) = 0 (k = 2^0 does not occur), a(1) = 1 (one value of k satisfying k <= 2^1), a(2) = 2 (two value of k satisfying k <= 2^2) and a(3) = 3 (three value of k satisfying k <= 2^3).
%Y Cf. A001223, A340948, A341531, A341532, A345332, A345333, A345334, A346776.
%K nonn,more
%O 0,3
%A _A.H.M. Smeets_, Aug 03 2021