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A345332
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.
5
0, 0, 0, 0, 1, 1, 6, 14, 28, 53, 122, 275, 597, 1203, 2456, 5111, 10573, 21662, 44553, 91246, 185422, 377264, 765956, 1552001, 3140326, 6349270, 12825847, 25891832
OFFSET
0,7
COMMENTS
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.
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).
FORMULA
a(n) = A340948(n) - (A345333(n) + A345334(n)).
CROSSREFS
KEYWORD
nonn,more
AUTHOR
A.H.M. Smeets, Jun 14 2021
STATUS
approved