A002386 Primes (lower end) with record gaps to the next consecutive prime: primes p(k) where p(k+1) - p(k) exceeds p(j+1) - p(j) for all j < k. (Formerly M0858 N0327)

2, 3, 7, 23, 89, 113, 523, 887, 1129, 1327, 9551, 15683, 19609, 31397, 155921, 360653, 370261, 492113, 1349533, 1357201, 2010733, 4652353, 17051707, 20831323, 47326693, 122164747, 189695659, 191912783, 387096133, 436273009, 1294268491

Offset

1,1

Comments

See the links by Jens Kruse Andersen et al. for very large gaps.

References

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 133.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 6.1, Table 1.

M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 14.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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 75 terms from M. F. Hasler and N. J. A. Sloane, terms n = 76..77 added by Charles R Greathouse IV)

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.

Jens Kruse Andersen and Norman Luhn, Record Prime Gaps

Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.

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 and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]

Thomas R. Nicely, New maximal prime gaps and first occurrences, Math. Comput. 68,227 (1999) 1311-1315.

Tomás Oliveira e Silva, Gaps between consecutive primes

D. Shanks, On maximal gaps between successive primes, Math. Comp., 18 (1964), 646-651.

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)

J. Young and A. Potler, First occurrence prime gaps, Math. Comp., 52 (1989), 221-224.

Index entries for primes, gaps between

Formula

a(n) = A000101(n) - A005250(n) = A008950(n-1) - 1. - M. F. Hasler, Dec 13 2007

A000720(a(n)) = A005669(n).

a(n) = A000040(A005669(n)). - M. F. Hasler, Apr 26 2014

Mathematica

s = {2}; gm = 1; Do[p = Prime[n]; g = Prime[n + 1] - p; If[g > gm, Print[p]; AppendTo[s, p]; gm = g], {n, 2, 1000000}]; s   (* Jean-François Alcover, Mar 31 2011 *)
Module[{nn=10^7, pr, df}, pr=Prime[Range[nn]]; df=Differences[pr]; DeleteDuplicates[ Thread[ {Most[ pr], df}], GreaterEqual[#1[[2]], #2[[2]]]&]][[All, 1]] (* The program generates the first 26 terms of the sequence. *) (* Harvey P. Dale, Sep 24 2022 *)

Prog

(PARI) a(n)=local(p, g); if(n<2, 2*(n>0), p=a(n-1); g=nextprime(p+1)-p; while(p=nextprime(p+1), if(nextprime(p+1)-p>g, break)); p) /* Michael Somos, Feb 07 2004 */
(PARI) p=q=2; g=0; until( g<(q=nextprime(1+p=q))-p && print1(q-g=q-p, ", "), ) \\ M. F. Hasler, Dec 13 2007

Crossrefs

Cf. A000040, A001223, A000101 (upper ends), A005250 (record gaps), A000230, A111870, A111943.

Cf. A005669, A134266, A070866.

See also A205827(n) = A000040(A214935(n)), A182514(n) = A000040(A241540(n)).

Sequence in context: A359292 A163834 A335366 * A000230 A256454 A133429

Adjacent sequences: A002383 A002384 A002385 * A002387 A002388 A002389

Keyword

nonn,nice,changed

Author

N. J. A. Sloane

Extensions

Definition clarified by Harvey P. Dale, Sep 24 2022

Status

approved