

A005250


Record gaps between primes.
(Formerly M0994)


56



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
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OFFSET

1,2


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 kth 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, SpringerVerlag, 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

John W. Nicholson, Table of n, a(n) for n = 1..77
Jens Kruse Andersen, The Top20 Prime Gaps
Jens Kruse Andersen, New record prime gap
Jens Kruse Andersen, Maximal 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 componentbycomponent construction of exactly integrating rank1 lattices and applications, arXiv:2012.14263 [math.NA], 2020.
A. Kourbatov, Maximal gaps between prime ktuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013 and J. Int. Seq. 16 (2013) #13.5.2.
A. 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.
A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015; and Int. Math. Forum, 10 (2015), 283288.
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), 6578.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv:1901.03785 [math.NT], 2019.
YaPing Lu and ShuFang Deng, An upper bound for the prime gap, arXiv:2007.15282 [math.GM], 2020.
T. R. Nicely, Some Results of Computational Research in Prime Numbers
T. R. Nicely, List of Gaps
Tomás Oliveira e Silva, Gaps between consecutive primes
D. Shanks, On maximal gaps between successive primes, Mathematics of Computation, 18(88), 646651. (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), 221224.
Index entries for primes, gaps between


FORMULA

a(n) = A000101(n)  A002386(n) = A008996(n1) + 1.  M. F. Hasler, Dec 13 2007
a(n+1) = 1 + Sum_{i=1..n} A053695(i).  John W. Nicholson, Jan 20 2016


MATHEMATICA

nn=10^7; Module[{d=Differences[Prime[Range[nn]]], ls={1}}, Table[If[d[[n]]> Last[ls], AppendTo[ls, d[[n]]]], {n, nn1}]; ls] (* Harvey P. Dale, Jul 23 2012 *)


PROG

(PARI) p=q=2; g=0; until( g<(q=nextprime(1+p=q))p & print1(g=qp, ", "), ) \\ M. F. Hasler, Dec 13 2007
(PARI) p=2; g=0; m=g; forprime(q=3, 10^13, g=qp; if(g>m, print(g", ", p, ", ", q); m=g); p=q) \\ John W. Nicholson, Dec 18 2016
(Haskell)
a005250 n = a005250_list !! (n1)
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


CROSSREFS

Records in A001223. For positions of records see A005669.
Cf. A000040, A002386, A000101, A008996, A058320, A107578.
Sequence in context: A274170 A173144 A049015 * A162762 A156097 A288793
Adjacent sequences: A005247 A005248 A005249 * A005251 A005252 A005253


KEYWORD

nonn,nice,changed


AUTHOR

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


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


STATUS

approved



