A005250 Record gaps between primes.

1, 2, 4, 6, 8, 14, 18, 20, 22, 34, 36, 44, 52, 72, 86, 96, 112, 114, 118, 132, 148, 154, 180, 210, 220, 222, 234, 248, 250, 282, 288, 292, 320, 336, 354, 382, 384, 394, 456, 464, 468, 474, 486, 490, 500, 514, 516, 532, 534, 540, 582, 588, 602, 652

Offset: 1

Author

N. J. A. Sloane, R. K. Guy, May 20 1991

Keywords

• nonn
• nice

Comments

Here a "gap" means prime(n+1) - prime(n), but in other references it can mean prime(n+1) - prime(n) - 1.

a(n+1)/a(n) <= 2, for all n <= 80, and a(n+1)/a(n) < 1 + f(n)/a(n) with f(n)/a(n) <= epsilon for some function f(n) and with 0 < epsilon <= 1. It also appears, with the small amount of data available, for all n <= 80, that a(n+1)/a(n) ~ 1. - John W. Nicholson, Jun 08 2014, updated Aug 05 2019

Equivalent to the above statement, A053695(n) = a(n+1) - a(n) <= a(n). - John W. Nicholson, Jan 20 2016

Conjecture: a(n) = O(n^2); specifically, a(n) <= n^2. - Alexei Kourbatov, Aug 05 2017

Conjecture: below the k-th prime, the number of maximal gaps is about 2*log(k), i.e., about twice as many as the expected number of records in a sequence of k i.i.d. random variables (see arXiv:1709.05508 for a heuristic explanation). - Alexei Kourbatov, Mar 16 2018

References

• B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 133.
• R. K. Guy, Unsolved Problems in Number Theory, A8.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Brian Kehrig, Table of n, a(n) for n = 1..83 (first 77 terms from John W. Nicholson, terms n=78..80 added by Rodolfo Ruiz-Huidobro)
• Jens Kruse Andersen and Norman Luhn, Record Prime Gaps
• Alex Beveridge, Table giving known values of A000101(n), A005250(n), A107578(n)
• R. P. Brent, J. H. Osborn and W. D. Smith, Lower bounds on maximal determinants of +-1 matrices via the probabilistic method, arXiv preprint arXiv:1211.3248 [math.CO], 2012.
• C. K. Caldwell, Table of prime gaps
• C. K. Caldwell, Gaps up to 1132
• R. K. Guy, Letter to N. J. A. Sloane, Aug 1986
• R. K. Guy, Letter to N. J. A. Sloane, 1987
• Lutz Kämmerer, A fast probabilistic component-by-component construction of exactly integrating rank-1 lattices and applications, arXiv:2012.14263 [math.NA], 2020.
• Alexei Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013 and J. Int. Seq. 16 (2013) #13.5.2.
• Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
• Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
• Alexei Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, arXiv:1506.03042 [math.NT], 2015; and J. Int. Seq. 18 (2015) #15.11.2.
• Alexei Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015; and Int. Math. Forum, 10 (2015), 283-288.
• Alexei Kourbatov, On the distribution of maximal gaps between primes in residue classes, arXiv preprint arXiv:1610.03340 [math.NT], 2016.
• Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression, arXiv:1709.05508 [math.NT], 2017; and Int. Math. Forum, 13 (2018), 65-78.
• Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv:1901.03785 [math.NT], 2019.
• Ya-Ping Lu and Shu-Fang Deng, An upper bound for the prime gap, arXiv:2007.15282 [math.GM], 2020.
• Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [See local copy in A007053]
• Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
• Tomás Oliveira e Silva, Gaps between consecutive primes
• D. Shanks, On maximal gaps between successive primes, Mathematics of Computation, 18(88), 646-651. (1964).
• Matt Visser, Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap, arXiv:1904.00499 [math.NT], 2019.
• Eric Weisstein's World of Mathematics, Prime Gaps
• Wikipedia, Prime gap
• Robert G. Wilson v, Notes (no date)
• Marek Wolf, A Note on the Andrica Conjecture, arXiv:1010.3945 [math.NT], 2010.
• J. Young and A. Potler, First occurrence prime gaps, Math. Comp., 52 (1989), 221-224.
• Index entries for primes, gaps between

Crossrefs

Records in A001223. For positions of records see A005669.

Cf. A000040, A002386, A000101, A008996, A058320, A107578.

Programs

Haskell

a005250 n = a005250_list !! (n-1)
a005250_list = f 0 a001223_list
   where f m (x:xs) = if x <= m then f m xs else x : f x xs
-- Reinhard Zumkeller, Dec 12 2012

Mathematica

nn=10^7;Module[{d=Differences[Prime[Range[nn]]],ls={1}},Table[If[d[[n]]> Last[ls],AppendTo[ls,d[[n]]]],{n,nn-1}];ls] (* Harvey P. Dale, Jul 23 2012 *)
DeleteDuplicates[Differences[Prime[Range[10^7]]],GreaterEqual] (* The program generates the first 26 terms of the sequence. *) (* Harvey P. Dale, May 12 2022 *)

PARI

p=q=2;g=0;until( g<(q=nextprime(1+p=q))-p & print1(g=q-p,","),) \\ M. F. Hasler, Dec 13 2007

PARI

p=2; g=0;m=g; forprime(q=3,10^13,g=q-p;if(g>m,print(g", ",p,", ",q);m=g);p=q) \\ John W. Nicholson, Dec 18 2016

Formula

a(n) = A000101(n) - A002386(n) = A008996(n-1) + 1. - M. F. Hasler, Dec 13 2007

a(n+1) = 1 + Sum_{i=1..n} A053695(i). - John W. Nicholson, Jan 20 2016

Extensions

More terms from Andreas Boerner (andreas.boerner(AT)altavista.net), Jul 11 2000

Additional comments from Frank Ellermann, Apr 20 2001

More terms from Robert G. Wilson v, Jan 03 2002, May 01 2006