A000101 Record gaps between primes (upper end) (compare A002386, which gives lower ends of these gaps).

3, 5, 11, 29, 97, 127, 541, 907, 1151, 1361, 9587, 15727, 19661, 31469, 156007, 360749, 370373, 492227, 1349651, 1357333, 2010881, 4652507, 17051887, 20831533, 47326913, 122164969, 189695893, 191913031, 387096383, 436273291, 1294268779

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

See A002386 for complete list of known terms and further references.

Except for a(1)=3 and a(2)=5, a(n) = A168421(k). Primes 3 and 5 are special in that they are the only primes which do not have a Ramanujan prime between them and their double, <= 6 and 10 respectively. Because of the large size of a gap, there are many repeats of the prime number in A168421. - John W. Nicholson, Dec 10 2013

References

• B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 133.
• 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 Alex Beveridge and M. F. Hasler, terms n=76..80 added by John W. Nicholson)
• Jens Kruse Andersen and Norman Luhn, Record Prime Gaps
• Alex Beveridge, Table giving known values of A000101(n), A005250(n), A107578(n)
• Andrew Booker, The Nth Prime Page
• Harald Cramer, On the order of magnitude of the difference between consecutive prime numbers, Acta Arith. 2 (1936), 396-403.
• Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and Terence Tao, Long gaps between primes, arXiv:1412.5029 [math.NT], 2014-2016.
• Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
• 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
• Thomas R. Nicely, First occurrence prime gaps [Local copy, pdf only]
• Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [See local copy in A007053]
• Tomás Oliveira e Silva, Computational projects
• Tomás Oliveira e Silva, Siegfried Herzog and Silvio Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18, Math. Comp., 83 (2014), 2033-2060.
• Daniel 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)
• 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

Cf. A000040, A001223 (differences between primes), A002386 (lower ends), A005250 (record gaps), A107578.

Cf. also A005669, A111943.

Programs

Mathematica

s = {3}; gm = 1; Do[p = Prime[n + 1]; g = p - Prime[n]; If[g > gm, Print[p]; AppendTo[s, p]; gm = g], {n, 2, 1000000}]; s  (* Jean-François Alcover, Mar 31 2011 *)

PARI

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

Formula

a(n) = A002386(n) + A005250(n) = A008995(n-1) + 1. - M. F. Hasler, Dec 13 2007