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A005250
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Increasing gaps between primes.
(Formerly M0994)
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42
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1, 2, 4, 6, 8, 14, 18, 20, 22, 34, 36, 44, 52, 72, 86, 96, 112, 114, 118, 132, 148, 154, 180, 210, 220, 222, 234, 248, 250, 282, 288, 292, 320, 336, 354, 382, 384, 394, 456, 464, 468, 474, 486, 490, 500, 514, 516, 532, 534, 540, 582, 588, 602, 652
(list;
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refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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Here a "gap" means prime(n+1)-prime(n), but in other references it can mean prime(n+1)-prime(n)-1.
a(n+1)/a(n) <= 2, for all n<=75, and a(n+1)/a(n) < 1 + f(n)/a(n) with f(n)/a(n) <= epsilon for some function f(n) and with 0 < epsilon <= 1. It also appears, with the small amount of data available, for all n<=75, that a(n+1)/a(n) ~ 1. - John W. Nicholson, Jun 08 2014
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 133.
R. K. Guy, Unsolved Problems in Number Theory, A8.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n=1..75
Jens Kruse Andersen, The Top-20 Prime Gaps
Jens Kruse Andersen, New record prime gap
Jens Kruse Andersen, Maximal gaps
Alex Beveridge, Table giving known values of A000101(n), A005250(n), A107578(n)
R. P. Brent, J. H. Osborn and W. D. Smith, Lower bounds on maximal determinants of +-1 matrices via the probabilistic method, arXiv preprint arXiv:1211.3248, 2012.
C. K. Caldwell, Table of prime gaps
C. K. Caldwell, Gaps up to 1132
A. Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242, 2013.
A. Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053, 2013
Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959, 2014
Alexei Kourbatov, Upper Bounds for Prime Gaps Related to Firoozbakht’s Conjecture, arXiv preprint, 2015.
T. R. Nicely, Some Results of Computational Research in Prime Numbers
T. R. Nicely, List of Gaps
Tomás Oliveira e Silva, Gaps between consecutive primes
D. Shanks, On maximal gaps between successive primes, Mathematics of Computation, 18(88), 646-651. (1964).
Marek Wolf, A Note on the Andrica Conjecture, arXiv:1010.3945.
J. Young and A. Potler, First occurrence prime gaps, Math. Comp., 52 (1989), 221-224.
Index entries for primes, gaps between
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FORMULA
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a(n) = A000101(n) - A002386(n) = A008996(n-1) + 1. - M. F. Hasler, Dec 13 2007
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MATHEMATICA
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a=0; s=""; For[i=1, i<10^5, p1=Prime[i]; p2=Prime[i+1]; e=p2-p1; If[e>a, s=s<>ToString[e]<>", "; a=e]; i++ ]; Print[s] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
nn=10^7; Module[{d=Differences[Prime[Range[nn]]], ls={1}}, Table[If[d[[n]]> Last[ls], AppendTo[ls, d[[n]]]], {n, nn-1}]; ls] (* Harvey P. Dale, Jul 23 2012 *)
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PROG
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(PARI) p=q=2; g=0; until( g<(q=nextprime(1+p=q))-p & print1(g=q-p, ", "), ) \\ M. F. Hasler, Dec 13 2007
(Haskell)
a005250 n = a005250_list !! (n-1)
a005250_list = f 0 a001223_list
where f m (x:xs) = if x <= m then f m xs else x : f x xs
-- Reinhard Zumkeller, Dec 12 2012
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CROSSREFS
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Records in A001223. For positions of records see A005669.
Cf. A002386, A000101, A008996, A058320, A107578.
Sequence in context: A089747 A173144 A049015 * A162762 A156097 A039597
Adjacent sequences: A005247 A005248 A005249 * A005251 A005252 A005253
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane, R. K. Guy, May 20 1991
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EXTENSIONS
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More terms from Andreas Boerner (andreas.boerner(AT)altavista.net), Jul 11 2000
Additional comments from Frank Ellermann, Apr 20 2001
More terms from Robert G. Wilson v, Jan 03 2002, May 01 2006
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STATUS
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approved
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