A341531 The number of even prime gaps g satisfying g == 2 (mod 6) out of the first 2^n even prime gaps.

1, 2, 3, 4, 7, 13, 23, 43, 85, 170, 320, 628, 1224, 2446, 4869, 9640, 19119, 37969, 75258, 149530, 297562, 592033, 1178763, 2348334, 4679406, 9326904, 18596999, 37086110, 73967842, 147557811, 294406743, 587477780, 1172420818, 2340067092, 4671002564

Offset

0,2

Comments

It seems that the fraction of prime gaps g, satisfying g == 2 (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.

Links

Martin Ehrenstein, Table of n, a(n) for n = 0..43

Formula

a(n) = 2^n - A340948(n) - A341532(n).

Prog

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

Crossrefs

Cf. A001223, A340948, A341532.

Sequence in context: A096723 A266498 A137495 * A099779 A270440 A000690

Adjacent sequences: A341528 A341529 A341530 * A341532 A341533 A341534

Keyword

nonn

Author

A.H.M. Smeets, Feb 13 2021

Extensions

a(29) and beyond from Martin Ehrenstein, Mar 01 2021

Status

approved