

A000101


Record gaps between primes (upper end) (compare A002386, which gives lower ends of these gaps).
(Formerly M2485 N0984)


78



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
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OFFSET

1,1


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, SpringerVerlag, 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

Alex Beveridge, M. F. Hasler, and John W. Nicholson, Table of n, a(n) for n = 1..80 [Added data from Thomas R. Nicely site.  John W. Nicholson, Oct 27 2021]
Jens Kruse Andersen, Maximal Prime Gaps
Alex Beveridge, Table giving known values of A000101(n), A005250(n), A107578(n)
Andrew Booker, The Nth Prime Page
H. Cramer, On the order of magnitude of the difference between consecutive prime numbers, Acta Arith. 2 (1936), 396403.
T. Oliveira e Silva, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18, Math. Comp., 83 (2014), 20332060.
Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, Terence Tao, Long gaps between primes, arXiv:1412.5029 [math.NT], 20142016.
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
D. Shanks, On maximal gaps between successive primes, Math. Comp., 18 (1964), 646651.
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) = A002386(n) + A005250(n) = A008995(n1) + 1.  M. F. Hasler, Dec 13 2007


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 (* JeanFrançois Alcover, Mar 31 2011 *)


PROG

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


CROSSREFS

Cf. A000040, A001223 (differences between primes), A002386 (lower ends), A005250 (record gaps), A107578.
Cf. also A005669, A111943.
Sequence in context: A057735 A095302 A335367 * A253899 A037152 A084748
Adjacent sequences: A000098 A000099 A000100 * A000102 A000103 A000104


KEYWORD

nonn,nice


AUTHOR

N. J. A. Sloane


STATUS

approved



