A053695 -id:A053695 - cp's OEIS frontend

A005250 Record gaps between primes.

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

Offset: 1

N. J. A. Sloane, R. K. Guy, May 20 1991

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 <= 80, 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 <= 80, that a(n+1)/a(n) ~ 1. - John W. Nicholson, Jun 08 2014, updated Aug 05 2019
Equivalent to the above statement, A053695(n) = a(n+1) - a(n) <= a(n). - John W. Nicholson, Jan 20 2016
Conjecture: a(n) = O(n^2); specifically, a(n) <= n^2. - Alexei Kourbatov, Aug 05 2017
Conjecture: below the k-th prime, the number of maximal gaps is about 2*log(k), i.e., about twice as many as the expected number of records in a sequence of k i.i.d. random variables (see arXiv:1709.05508 for a heuristic explanation). - Alexei Kourbatov, Mar 16 2018

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).

Brian Kehrig, Table of n, a(n) for n = 1..83 (first 77 terms from John W. Nicholson, terms n=78..80 added by Rodolfo Ruiz-Huidobro)
Jens Kruse Andersen and Norman Luhn, Record Prime 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 [math.CO], 2012.
C. K. Caldwell, Table of prime gaps
C. K. Caldwell, Gaps up to 1132
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.
Alexei Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013 and J. Int. Seq. 16 (2013) #13.5.2.
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
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, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015; and Int. Math. Forum, 10 (2015), 283-288.
Alexei Kourbatov, On the distribution of maximal gaps between primes in residue classes, arXiv preprint arXiv:1610.03340 [math.NT], 2016.
Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression, arXiv:1709.05508 [math.NT], 2017; and Int. Math. Forum, 13 (2018), 65-78.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv:1901.03785 [math.NT], 2019.
Ya-Ping Lu and Shu-Fang Deng, An upper bound for the prime gap, arXiv:2007.15282 [math.GM], 2020.
Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [See local copy in A007053]
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Tomás Oliveira e Silva, Gaps between consecutive primes
D. Shanks, On maximal gaps between successive primes, Mathematics of Computation, 18(88), 646-651. (1964).
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

Records in A001223. For positions of records see A005669.

Cf. A000040, A002386, A000101, A008996, A058320, A107578.

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

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 *)
DeleteDuplicates[Differences[Prime[Range[10^7]]],GreaterEqual] (* The program generates the first 26 terms of the sequence. *) (* Harvey P. Dale, May 12 2022 *)

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

p=2; g=0;m=g; forprime(q=3,10^13,g=q-p;if(g>m,print(g", ",p,", ",q);m=g);p=q) \\ John W. Nicholson, Dec 18 2016

a(n) = A000101(n) - A002386(n) = A008996(n-1) + 1. - M. F. Hasler, Dec 13 2007

a(n+1) = 1 + Sum_{i=1..n} A053695(i). - John W. Nicholson, Jan 20 2016

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

A053686 Record gaps between consecutive primes that repeat at least once before a new record occurs.

Original entry on oeis.org

2, 4, 6, 14, 34, 36, 52, 86, 132, 154, 250, 336

Offset: 1

Jeff Burch, Mar 23 2000

Scan the sequence of prime differences (A001223) looking for new records, but append the record difference to the present sequence only if the difference appears at least twice in A001223 before it is beaten by a new record. - N. J. A. Sloane, Dec 30 2007
The sequence of primes where these gaps first appear is A133788.
These are the numbers that appear two or more times in A085237. - David W. Wilson, Dec 31 2007

Jens Kruse Andersen, Maximal gaps

Cf. A001223, A053695, A133788, A085237, A002386.

More terms from Naohiro Nomoto, Jul 23 2001
Corrected by Jorge Coveiro, Jul 24 2006
More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Sep 13 2006
There were still two erroneous terms. The terms a(1) - a(11) now shown have been verified by Farideh Firoozbakht, Dec 31 2007. Edited by N. J. A. Sloane, Jan 30 2008.
a(12) from Donovan Johnson, Nov 24 2008

A104138 Smallest prime followed by n or more composites.

Original entry on oeis.org

2, 3, 7, 7, 23, 23, 89, 89, 113, 113, 113, 113, 113, 113, 523, 523, 523, 523, 887, 887, 1129, 1129, 1327, 1327, 1327, 1327, 1327, 1327, 1327, 1327, 1327, 1327, 1327, 1327, 9551, 9551, 15683, 15683, 15683, 15683, 15683, 15683, 15683, 15683, 19609

Offset: 0

Lekraj Beedassy, Mar 07 2005

nonn

Except for a(1), records occur at even values of n, and each term appears an even number of times consecutively. (Proof. A maximal run of composites must begin and end at even numbers.) - Jonathan Sondow, May 31 2014

			a(10)=113 because it is the first prime occurring before primes 199,211,293,317,467,509,... all followed by at least ten successive composites.

Charles R Greathouse IV, Table of n, a(n) for n = 0..1000

Cf. A002386, A005250, A030296, A053695.

Record prime A002386(n+1) appears A053695(n-1) times, for n>1.

a(n) = A030296(n) - 1, for n > 0. - Jonathan Sondow, May 31 2014

a(34) corrected by Charles R Greathouse IV, Aug 09 2011

A307325 a(n) is the smallest number k for which prime(k+1) - prime(k) is greater than n.

Original entry on oeis.org

2, 4, 4, 9, 9, 24, 24, 30, 30, 30, 30, 30, 30, 99, 99, 99, 99, 154, 154, 189, 189, 217, 217, 217, 217, 217, 217, 217, 217, 217, 217, 217, 217, 1183, 1183, 1831, 1831, 1831, 1831, 1831, 1831, 1831, 1831, 2225, 2225, 2225, 2225, 2225, 2225, 2225, 2225, 3385, 3385, 3385, 3385

Offset: 1

Marius A. Burtea, Apr 02 2019

nonn

For any n there is an infinity of numbers m for which prime(m+1) - prime(m) is greater than n.

It appears that the sequence of lengths of successive runs is equal to A053695. - Marc Bofill Janer, May 21 2019

			For n = 2, prime(2) - prime(1) = 3 - 2 = 1, prime(3) - prime(2) = 5 - 3 = 2, prime(5) - prime(4) = 11 - 7 = 4, so a(2) = 4.

Laurențiu Panaitopol, Dinu Șerbănescu, Number theory and combinatorial problems for juniors, Ed.Gil, Zalău, (2003), ch. 1, p.7, pr. 25. (in Romanian).

Cf. A000040, A001223, A005250, A005669.

Cf. A053695, A144309.

v=primes(1000000);
for u=1:100; ss=1;
    while and(v(ss+1)-v(ss)<=u,ss

v:=PrimesUpTo(10000000);
sol:=[];
for u in [1..60] do
   for ss in [1..#v-1] do
    if v[ss+1]-v[ss] gt u then
         sol[u]:=ss;
         break;
     end if;
   end for;
end for;
   sol;

a(n) = my(k=1); while(prime(k+1) - prime(k) <= n, k++); k; \\ Michel Marcus, Apr 03 2019

a(2*n) = a(2*n+1) = A144309(n+1) for n>=1. - Georg Fischer, Dec 05 2022

cp's OEIS Frontend

A243000 max {A053695(k) | k < n+2} - max {A053695(k) | k < n+1}.

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A270878 Record differences between record prime gaps, that is, terms of A053695 that set a record.

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A005250 Record gaps between primes.

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A053686 Record gaps between consecutive primes that repeat at least once before a new record occurs.

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A104138 Smallest prime followed by n or more composites.

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A307325 a(n) is the smallest number k for which prime(k+1) - prime(k) is greater than n.

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