cp's OEIS Frontend

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

%I A000101 M2485 N0984 #160 Feb 16 2025 08:32:19
%S A000101 3,5,11,29,97,127,541,907,1151,1361,9587,15727,19661,31469,156007,
%T A000101 360749,370373,492227,1349651,1357333,2010881,4652507,17051887,
%U A000101 20831533,47326913,122164969,189695893,191913031,387096383,436273291,1294268779
%N A000101 Record gaps between primes (upper end) (compare A002386, which gives lower ends of these gaps).
%C A000101 See A002386 for complete list of known terms and further references.
%C A000101 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
%D A000101 B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 133.
%D A000101 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000101 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000101 Brian Kehrig, <a href="/A000101/b000101.txt">Table of n, a(n) for n = 1..83</a> (first 75 terms from Alex Beveridge and M. F. Hasler, terms n=76..80 added by John W. Nicholson)
%H A000101 Jens Kruse Andersen and Norman Luhn, <a href="https://www.pzktupel.de/JensKruseAndersen/risinggap.php">Record Prime Gaps</a>
%H A000101 Alex Beveridge, <a href="/A000101/a000101_1.txt">Table giving known values of A000101(n), A005250(n), A107578(n)</a>
%H A000101 Andrew Booker, <a href="https://t5k.org/nthprime/">The Nth Prime Page</a>
%H A000101 Harald Cramer, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa2/aa212.pdf">On the order of magnitude of the difference between consecutive prime numbers</a>, Acta Arith. 2 (1936), 396-403.
%H A000101 Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and Terence Tao, <a href="http://arxiv.org/abs/1412.5029">Long gaps between primes</a>, arXiv:1412.5029 [math.NT], 2014-2016.
%H A000101 Alexei Kourbatov, <a href="http://arxiv.org/abs/1309.4053">Tables of record gaps between prime constellations</a>, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
%H A000101 Alexei Kourbatov and Marek Wolf, <a href="https://arxiv.org/abs/1901.03785">Predicting maximal gaps in sets of primes</a>, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
%H A000101 Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html"> First occurrence prime gaps</a>
%H A000101 Thomas R. Nicely, <a href="/A000101/a000101.pdf">First occurrence prime gaps</a> [Local copy, pdf only]
%H A000101 Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/index.html">Some Results of Computational Research in Prime Numbers</a> [See local copy in A007053]
%H A000101 Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/hobbies.html">Computational projects</a>
%H A000101 Tomás Oliveira e Silva, Siegfried Herzog and Silvio Pardi, <a href="https://doi.org/10.1090/S0025-5718-2013-02787-1">Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18</a>, Math. Comp., 83 (2014), 2033-2060.
%H A000101 Daniel Shanks, <a href="http://www.jstor.org/stable/2002951">On maximal gaps between successive primes</a>, Math. Comp., 18 (1964), 646-651.
%H A000101 Matt Visser, <a href="https://arxiv.org/abs/1904.00499">Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap</a>, arXiv:1904.00499 [math.NT], 2019.
%H A000101 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeGaps.html">Prime Gaps</a>
%H A000101 Wikipedia, <a href="https://en.wikipedia.org/wiki/Prime_gap">Prime gap</a>
%H A000101 Robert G. Wilson v, <a href="/A005250/a005250.pdf">Notes (no date)</a>
%H A000101 Marek Wolf, <a href="http://arxiv.org/abs/1010.3945">A Note on the Andrica Conjecture</a>, arXiv:1010.3945 [math.NT], 2010.
%H A000101 J. Young and A. Potler, <a href="http://www.jstor.org/stable/2008665">First occurrence prime gaps</a>, Math. Comp., 52 (1989), 221-224.
%H A000101 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%F A000101 a(n) = A002386(n) + A005250(n) = A008995(n-1) + 1. - _M. F. Hasler_, Dec 13 2007
%t A000101 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 *)
%o A000101 (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
%Y A000101 Cf. A000040, A001223 (differences between primes), A002386 (lower ends), A005250 (record gaps), A107578.
%Y A000101 Cf. also A005669, A111943.
%K A000101 nonn,nice
%O A000101 1,1
%A A000101 _N. J. A. Sloane_