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A340948
The number of even prime gaps g, satisfying g == 0 (mod 6), out of the first 2^n even prime gaps.
8
0, 0, 0, 1, 4, 8, 19, 44, 88, 173, 385, 793, 1649, 3301, 6647, 13490, 27299, 55136, 111630, 225230, 453453, 913088, 1836779, 3691941, 7418406, 14900625, 29914868, 60045509, 120499773, 241755292, 484928340, 972528090, 1950125661, 3909800410, 7837864058
OFFSET
0,5
COMMENTS
It seems that the fraction of prime gaps g, satisfying g == 0 (mod 6), tends to a constant, say c, when the number of prime gaps tends to infinity. From n = 43 we obtain that c > 0.463, while it can be argued heuristically that c < 0.5.
Meanwhile, the fractions of prime gaps g, satisfying either g == 2 (mod 6) or g == 4 (mod 6), seem to tend both to another constant, (1-c)/2, when the number of prime gaps tends to infinity (see A341531 and A341532).
LINKS
FORMULA
a(n) = 2^n - A341531(n) - A341532(n). - Martin Ehrenstein, Mar 01 2021
a(n) = A345332(n) + A345333(n) + A345334(n). - A.H.M. Smeets, Jul 10 2021
EXAMPLE
The first even prime gaps are given by 2, 2, 4, 2,..., A001223 without the leading 1.
The 8th even prime gap is the first gap satisfying g == 0 (mod 6), so a(3) = 1.
PROG
(PARI) a(n) = my(vp=primes(2^n+2)); #select(x->!(x%6), vector(#vp-1, k, vp[k+1]-vp[k])); \\ Michel Marcus, Feb 04 2021
KEYWORD
nonn
AUTHOR
A.H.M. Smeets, Jan 31 2021
EXTENSIONS
a(26) corrected by Martin Ehrenstein, Feb 14 2021
a(29) and beyond from Martin Ehrenstein, Feb 18 2021
STATUS
approved