A166597 Let p = largest prime <= n, with p(0)=p(1)=0, and let q = smallest prime > n; then a(n) = q-p.

2, 2, 1, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 2, 2, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 2, 2, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 2, 2, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 4, 4, 4, 4, 2, 2, 4, 4

Offset: 0

Author

Daniel Forgues, Oct 17 2009

Keywords

• nonn

Comments

Note the large prime gap of 72 between 31397 and 31469. This is the prime gap with the largest merit (cf. A111870), 72/log(31397)=6.95352 for primes less than 100000. Also 72/(log(31397))^2=0.67154 (cf. conjectures of Cramer-Granville, Shanks and Wolf) is largest for primes less than 100000. - Daniel Forgues, Oct 23 2009

Examples

			a(0) = 2 since the least prime greater than 0 is 2 (gap of 2 from 0 to 2).
a(9) = 4 since the least prime greater than 9 is 11 (gap of 4 from 7 to 11).
a(11) = 2 since the least prime greater than 11 is 13 (gap of 2 from 11 to 13).
		

Links

• Daniel Forgues, Table of n, a(n) for n = 0..100000
• Eric Weisstein's World of Mathematics, Prime Gaps.
• Eric Weisstein's World of Mathematics, Cramer-Granville Conjecture.
• Eric Weisstein's World of Mathematics, Shanks Conjecture.

Crossrefs

Cf. A151800, A166594.

Cf. A111870. - Daniel Forgues, Oct 23 2009

See A327441 for the classic G(n) version. - N. J. A. Sloane, Sep 11 2019

Cf. A001223, A000720, A007917, A007918, A151799.

Programs

Maple

2,2,seq(nextprime(n)-prevprime(n+1), n=2..100); # Ridouane Oudra, Dec 28 2024

Mathematica

f[n_]:=Module[{a=If[PrimeQ[n],n,NextPrime[n,-1]]}, NextPrime[n]-a]; Join[{2,2},Array[f,120,2]] (* Harvey P. Dale, May 17 2011 *)

PARI

a(n) = nextprime(n+1) - precprime(n); \\ Michel Marcus, Mar 02 2023

Formula

From Ridouane Oudra, Dec 28 2024: (Start)

a(n) = A001223(A000720(n)), for n>1.

a(n) = A151800(n) - A007917(n), for n>1.

a(n) = A007918(n+1) - A151799(n+1), for n>1. (End)

Extensions

Definition rephrased by N. J. A. Sloane, Oct 25 2009