A340948 The number of even prime gaps g, satisfying g == 0 (mod 6), out of the first 2^n even prime gaps.

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

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

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

Crossrefs

Cf. A001223, A270190, A341531, A341532, A345332, A345333, A345334.

Sequence in context: A083579 A335714 A215112 * A265108 A328184 A332367

Adjacent sequences: A340945 A340946 A340947 * A340949 A340950 A340951

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