%I #18 Mar 02 2023 11:24:24

%S 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,

%T 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,

%U 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

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

%C 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

%H Daniel Forgues, <a href="/A166597/b166597.txt">Table of n, a(n) for n = 0..100000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeGaps.html">Prime Gaps</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cramer-GranvilleConjecture.html">Cramer-Granville Conjecture</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ShanksConjecture.html">Shanks Conjecture</a>.

%e a(0) = 2 since the least prime greater than 0 is 2 (gap of 2 from 0 to 2).

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

%e a(11) = 2 since the least prime greater than 11 is 13 (gap of 2 from 11 to 13).

%t 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 *)

%o (PARI) a(n) = nextprime(n+1) - precprime(n); \\ _Michel Marcus_, Mar 02 2023

%Y Cf. A151800, A166594.

%Y Cf. A111870. - _Daniel Forgues_, Oct 23 2009

%Y See A327441 for the classic G(n) version. - _N. J. A. Sloane_, Sep 11 2019

%K nonn

%O 0,1

%A _Daniel Forgues_, Oct 17 2009

%E Definition rephrased by _N. J. A. Sloane_, Oct 25 2009