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The number of even prime gaps g, satisfying g == 4 (mod 6), out of the first 2^n even prime gaps.
8

%I #13 Mar 06 2021 10:46:23

%S 0,0,1,3,5,11,22,41,83,169,319,627,1223,2445,4868,9638,19118,37967,

%T 75256,149528,297561,592031,1178762,2348333,4679404,9326903,18596997,

%U 37086109,73967841,147557809,294406741,587477778,1172420817,2340067090,4671002562

%N The number of even prime gaps g, satisfying g == 4 (mod 6), out of the first 2^n even prime gaps.

%C It seems that the fraction of prime gaps g, satisfying g == 4 (mod 6), tends to a constant, say c, when the number of prime gaps tends to infinity. From n = 28 we obtain that c < 0.276, while it can be argued heuristically that c > 0.25.

%H Martin Ehrenstein, <a href="/A341532/b341532.txt">Table of n, a(n) for n = 0..43</a>

%F a(n) = 2^n - A340948(n) - A341531(n).

%o (PARI) a(n) = my(vp=primes(2^n+2)); #select(x->((x%6)==4), vector(#vp-1, k, vp[k+1]-vp[k])); \\ _Michel Marcus_, Feb 16 2021

%Y Cf. A001223, A340948, A341531.

%K nonn

%O 0,4

%A _A.H.M. Smeets_, Feb 13 2021

%E a(29) and beyond from _Martin Ehrenstein_, Mar 01 2021