A111943 -id:A111943 - cp's OEIS frontend

A002386 Primes (lower end) with record gaps to the next consecutive prime: primes p(k) where p(k+1) - p(k) exceeds p(j+1) - p(j) for all j < k.

2, 3, 7, 23, 89, 113, 523, 887, 1129, 1327, 9551, 15683, 19609, 31397, 155921, 360653, 370261, 492113, 1349533, 1357201, 2010733, 4652353, 17051707, 20831323, 47326693, 122164747, 189695659, 191912783, 387096133, 436273009, 1294268491

Offset: 1

N. J. A. Sloane

See the links by Jens Kruse Andersen et al. for very large gaps.

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 133.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 6.1, Table 1.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 14.
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).

Brian Kehrig, Table of n, a(n) for n = 1..83 (first 75 terms from M. F. Hasler and N. J. A. Sloane, terms n = 76..77 added by Charles R Greathouse IV)
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.
Jens Kruse Andersen and Norman Luhn, Record Prime Gaps
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
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 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 [For local copy see A000101]
Thomas R. Nicely, New maximal prime gaps and first occurrences, Math. Comput. 68,227 (1999) 1311-1315.
Tomás Oliveira e Silva, Gaps between consecutive primes
D. 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)
J. Young and A. Potler, First occurrence prime gaps, Math. Comp., 52 (1989), 221-224.
Index entries for primes, gaps between

Cf. A000040, A001223, A000101 (upper ends), A005250 (record gaps), A000230, A111870, A111943.
Cf. A005669, A134266, A070866.
See also A205827(n) = A000040(A214935(n)), A182514(n) = A000040(A241540(n)).

s = {2}; gm = 1; Do[p = Prime[n]; g = Prime[n + 1] - p; If[g > gm, Print[p]; AppendTo[s, p]; gm = g], {n, 2, 1000000}]; s   (* Jean-François Alcover, Mar 31 2011 *)
Module[{nn=10^7,pr,df},pr=Prime[Range[nn]];df=Differences[pr];DeleteDuplicates[ Thread[ {Most[ pr],df}],GreaterEqual[#1[[2]],#2[[2]]]&]][[All,1]] (* The program generates the first 26 terms of the sequence. *) (* Harvey P. Dale, Sep 24 2022 *)

a(n)=local(p,g);if(n<2,2*(n>0),p=a(n-1);g=nextprime(p+1)-p;while(p=nextprime(p+1),if(nextprime(p+1)-p>g,break));p) /* Michael Somos, Feb 07 2004 */

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

a(n) = A000101(n) - A005250(n) = A008950(n-1) - 1. - M. F. Hasler, Dec 13 2007
A000720(a(n)) = A005669(n).
a(n) = A000040(A005669(n)). - M. F. Hasler, Apr 26 2014

Definition clarified by Harvey P. Dale, Sep 24 2022

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

Original entry on oeis.org

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

N. J. A. Sloane

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

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).

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

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

Cf. also A005669, A111943.

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 *)

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

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

A111870 Prime p with prime gap q - p of n-th record merit, where q is smallest prime larger than p and the merit of a prime gap is (q-p)/log(p).

Original entry on oeis.org

2, 3, 7, 113, 1129, 1327, 19609, 31397, 155921, 360653, 370261, 1357201, 2010733, 17051707, 20831323, 191912783, 436273009, 2300942549, 3842610773, 4302407359, 10726904659, 25056082087, 304599508537, 461690510011, 1346294310749, 1408695493609, 1968188556461, 2614941710599, 13829048559701, 19581334192423, 218209405436543, 1693182318746371

Offset: 1

N. J. A. Sloane, based on correspondence with Ed Pegg Jr, Nov 23 2005

nonn

As I understand it, the sequence refers to "Smallest prime p whose following gap has bigger merit than the other primes smaller than p." If that is the case, then it has an error. The sequence starts: 2, 3, 7, 113, 1129, 1327, 19609, 31397, 155921, 360653, 370261, 1357201, 4652353, 2010733, ... but you can see that 4652353 > 2010733, so in any case it should be listed after, not before it. But above that, its merit is 10.03 < 10.20, the merit of 2010733, so it is not in a mistaken position: it shouldn't appear in the sequence. - Jose Brox, Dec 31 2005
The logarithmic (base 10) graph seems to be linearly asymptotic to n with slope ~ 1/log(10) which would imply that: log(prime with n-th record merit) ~ n as n goes to infinity. - N. J. A. Sloane, Aug 27 2010
The sequence b(n) = (prime(n+1)/prime(n))^n is increasing for terms prime(n) of this sequence. - Thomas Ordowski, May 04 2012
The smallest prime(n) such that (prime(n+1)/prime(n))^n is increasing: 2, 3, 7, 23, 113, 1129, 1327, ... (A205827). - Thomas Ordowski, May 04 2012
(prime(n+1)/prime(n))^n > 1 + merit(n) for n > 2, where merit(n) = (prime(n+1)-prime(n))/log(prime(n)). - Thomas Ordowski, May 14 2012
Merit(1) + merit(2) + ... + merit(n) =: S(n) ~ n, where merit(n) is as above. - Thomas Ordowski, Aug 03 2012
For the index of a(n), see the comment at A214935. - John W. Nicholson, Nov 21 2013

			The first few entries correspond to the following gaps. The table gives n, p, gap = q-p and the merit of the gap.
   1,       2,   1, 1.4427
   2,       3,   2, 1.82048
   3,       7,   4, 2.05559
   4,     113,  14, 2.96147
   5,    1129,  22, 3.12985
   6,    1327,  34, 4.72835
   7,   19609,  52, 5.26116
   8,   31397,  72, 6.95352
   9,  155921,  86, 7.19238
  10,  360653,  96, 7.50254
  11,  370261, 112, 8.73501
  12, 1357201, 132, 9.34782

Ed Pegg, Jr., Posting to Seq Fan mailing list, Nov 23 2005

Hugo Pfoertner, Table of n, a(n) for n = 1..39, extracted from pzktupel link.
Jens Kruse Andersen, The Top-20 Prime Gaps
Jens Kruse Andersen, Maximal gaps
Jens Kruse Andersen, Record prime gaps, current version maintained by Norman Luhn.
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Eric Weisstein's World of Mathematics, Prime Gaps

For the gaps, see A111871.

Cf. A002386, A111943, A214935, A277552.

With[{s = Map[(#2 - #1)/Log[#1] & @@ # &, Partition[Prime@ Range[10^6], 2, 1]]}, Map[Prime@ FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Jul 19 2018 *)

a(n) = A277552(n) - A111871(n). - Bobby Jacobs, Nov 13 2016

Corrected by Jose Brox, Dec 31 2005
Corrected and edited by Daniel Forgues, Oct 23 2009
Further edited by Daniel Forgues, Nov 01 2009, Nov 13 2009, Nov 24 2009

A166363 Number of primes in the half-open interval (n(log(n))^2..(n+1)(log(n+1))^2].

Original entry on oeis.org

0, 2, 2, 1, 3, 1, 2, 3, 2, 2, 3, 2, 2, 4, 1, 2, 3, 3, 3, 3, 2, 2, 5, 2, 3, 4, 1, 3, 3, 3, 4, 3, 3, 3, 4, 3, 3, 4, 1, 3, 3, 5, 3, 4, 4, 3, 3, 3, 4, 3, 3, 4, 4, 4, 2, 3, 4, 3, 3, 4, 5, 3, 5, 4, 2, 3, 3, 6, 2, 4, 5, 3, 2, 2, 3, 6, 3, 6, 3, 4, 4, 6, 3, 4, 3, 4, 4, 4, 2, 3, 6, 3, 3, 2, 6, 5, 2, 6, 3, 5, 3, 2, 5, 4, 4

Offset: 1

Daniel Forgues, Oct 12 2009

nonn

The open-closed half-open intervals form a partition of the real line for x > 0, thus each prime appears in a unique interval.
The n-th interval length is ~ (log(n+1/2))^2 + 2*log(n+1/2) ~ (log(n))^2 as n goes to infinity.
The n-th interval prime density is ~ 1/[log(n+1/2)+2*log(log(n+1/2))] ~ 1/log(n) as n goes to infinity.
The expected number of primes in the n-th interval is ~ [(log(n+1/2))^2 + 2*log(n+1/2)] / [log(n+1/2)+2*log(log(n+1/2))] ~ log(n) as n goes to infinity.
For n = 1 there is no prime.
If it can be proved that each interval always contains at least one prime, this would constitute even shorter intervals than A166332(n), let alone A143898(n), as n gets large.
The Shanks Conjecture and the Cramer-Granville Conjecture tell us that the intervals of length (log(n))^2 are of very critical length (the constant M > 1 of the Cramer-Granville Conjecture definitely matters!). There seems to be some risk that one such interval does not contain a prime.
The Wolf Conjecture (which agrees better with numerical evidence) seems more in favor of each interval's containing at least one prime.
From Charles R Greathouse IV, May 13 2010: (Start)
Not all intervals > 1 contain primes!
a(n) = 0 for n = 1, 4977, 17512, 147127, 76082969 (and no others up to 10^8).
Higher values include 731197850, 2961721173, 2103052050563, 188781483769833, 1183136231564246 but this list is not exhaustive.
The intervals have length (log n)^2 + 2*log n + o(1). In the Cramer model, the probability that a given integer in the interval would be prime is approximately 1/(log n + 2*log log n). Tedious calculation gives the probability that a(n) = 0 in the Cramer model as 3C(log n)^2/n * (1 + o(1)) with C = exp(-5/2)/3. Thus under that model we would expect to find roughly C*(log N)^3 numbers n up to N with a(n) = 0. In fact, the numbers are not that common since the probabilities are not independent.
(End)
The similar sequence A345755 relies on intervals that are slightly more than twice as wide as those in the present sequence. A345755 does not include zero entries for n <= 2772, suggesting that the lengths of prime gaps may be bracketed by the two sequences. We conjecture that prime gaps may be larger than log(p)^2, but are not larger than log_2(p)^2. - Hal M. Switkay, Aug 29 2023

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Gaps.
Eric Weisstein's World of Mathematics, Cramer-Granville Conjecture.
Eric Weisstein's World of Mathematics, Shanks Conjecture (and Wolf Conjecture.)

Cf. A166332, A000720, A111943, A143898, A134034, A143935, A144140 (primes between successive n^K, for different K), A014085 (primes between successive squares).

Cf. A182315, A345755.

a(n)=sum(k=ceil(n*log(n)^2),floor((n+1)*log(n+1)^2), isprime(k)) \\ Charles R Greathouse IV, Aug 21 2015

a(n) = pi((n+1)*(log(n+1))^2) - pi(n*(log(n))^2) since the intervals are half-open properly.

Edited by Daniel Forgues, Oct 18 2009 and Nov 01 2009

Edited by Charles R Greathouse IV, May 13 2010

A211073 Primes p followed by a gap of at least 1/2 * log(p)^2.

Original entry on oeis.org

2, 3, 5, 7, 13, 23, 31, 113, 1327, 19609, 25471, 31397, 34061, 43331, 44293, 155921, 188029, 212701, 265621, 338033, 360653, 370261, 396733, 404851, 492113, 544279, 576791, 604073, 838249, 860143, 1098847, 1139993, 1313467, 1349533, 1357201, 1388483, 1444309

Offset: 1

Charles R Greathouse IV, Apr 26 2012

nonn

Primes followed by unusually long prime gaps.

The Cramér model suggests that there are about 2*sqrt(x/log^2 x) elements up to x. - Charles R Greathouse IV, Mar 18 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002386, A082885, A111943, A182315.

Select[Prime[Range[10^4]], NextPrime[#] - # > (Log[#]^2)/2 &] (* Alonso del Arte, Jun 02 2013 *)

G=1; p=2; forprime(q=3, 1e7, if(q-p>=G && q-p>log(p)^2/2, G=ceil(log(p)^2/2); print1(p", ")); p=q)

Primes p such that all integers in (p, p + 0.5 * log(p)^2) are composite.

A241540 Indices of primes p in A182514, i.e., a(n) = primepi(p) = A000720(A182514(n)).

Original entry on oeis.org

1, 2, 4, 30, 217, 49749629143526

Offset: 1

M. F. Hasler, Apr 25 2014

a(6) = A214935(33) = A000720(A205827(33)).

Cf. A053302, A053976, A073861, A111870, A111943, A182514.

See also A214935(n)=A000720(A205827(n)), A005669(n) = A000720(A002386(n)).

A211075 Primes p followed by prime gap of length log(p/log(p)^2)^2.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 103, 109, 113, 127, 131, 139, 151, 157, 163, 167, 173, 181, 193, 199, 211, 233, 241, 251, 257, 263, 271, 283, 293, 317, 331, 337, 353, 359, 367, 373, 383, 389, 401, 409

Offset: 1

Charles R Greathouse IV, May 06 2012

nonn

Primes followed by unusually long prime gaps.

Charles R Greathouse IV, Table of n, a(n) for n = 1..4139

Cf. A211073, A002386, A082885, A111943, A182315.

Select[Prime[Range[200]], NextPrime[#] - # > Log[#/Log[#]^2]^2 &] (* Alonso del Arte, Jun 02 2013 *)

G=1;p=3;forprime(q=5,1e7,if(q-p>=G,G=log(p/log(p)^2)^2; if(q-p>=G, print1(p", ")));p=q)

cp's OEIS Frontend

A002386 Primes (lower end) with record gaps to the next consecutive prime: primes p(k) where p(k+1) - p(k) exceeds p(j+1) - p(j) for all j < k.

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

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A111870 Prime p with prime gap q - p of n-th record merit, where q is smallest prime larger than p and the merit of a prime gap is (q-p)/log(p).

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A166363 Number of primes in the half-open interval (n*(log(n))^2..(n+1)*(log(n+1))^2].

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A211073 Primes p followed by a gap of at least 1/2 * log(p)^2.

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A241540 Indices of primes p in A182514, i.e., a(n) = primepi(p) = A000720(A182514(n)).

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A211075 Primes p followed by prime gap of length log(p/log(p)^2)^2.

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A166363 Number of primes in the half-open interval (n(log(n))^2..(n+1)(log(n+1))^2].