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.

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

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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

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)).

Programs

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 *)

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

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

Extensions

Definition clarified by Harvey P. Dale, Sep 24 2022