A008995 -id:A008995 - cp's OEIS frontend

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

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

A008996 Increasing length runs of consecutive composite numbers (records).

Original entry on oeis.org

1, 3, 5, 7, 13, 17, 19, 21, 33, 35, 43, 51, 71, 85, 95, 111, 113, 117, 131, 147, 153, 179, 209, 219, 221, 233, 247, 249, 281, 287, 291, 319, 335, 353, 381, 383, 393, 455, 463, 467, 473, 485, 489, 499, 513, 515, 531, 533, 539, 581, 587, 601, 651, 673, 715, 765

Offset: 1

Mark Cramer (m.cramer(AT)qut.edu.au), Mar 15 1996

Conjecture: a(n) = O(n^2); specifically, a(n) <= n^2. - Alexei Kourbatov, Jan 23 2019

Bert Sierra, Table of n, a(n) for n = 1..82 (derived from A005250; first 74 terms from Jens Kruse Andersen)
Jens Kruse Andersen, Maximal Prime Gaps
Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression, arXiv:1709.05508 [math.NT], 2017; Int. Math. Forum, 13 (2018), 65-78.
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.
Eric Weisstein's World of Mathematics, Prime Gaps
Index entries for primes, gaps between

Cf. A005250, A008950, A008995, A058320.

Cf. A010051, A046933.

a008996 n = a008996_list !! (n-1)
a008996_list = 1 : f 0 (filter (> 1) $
                        map length $ group $ drop 3 a010051_list)
   where f m (u : us) = if u <= m then f m us else u : f u us
-- Reinhard Zumkeller, Nov 27 2012

maxGap = 1; Reap[ Do[ gap = Prime[n+1] - Prime[n]; If[gap > maxGap, Print[gap-1]; Sow[gap-1]; maxGap = gap], {n, 2, 10^8}]][[2, 1]] (* Jean-François Alcover, Jun 12 2013 *)
Module[{nn=10^8,cmps},cmps=Table[If[CompositeQ[n],1,{}],{n,nn}];DeleteDuplicates[ Rest[ Length/@ Split[cmps]],GreaterEqual]] (* The program generates the first 24 terms of the sequnece. To generate more, increase the nn constant. *) (* Harvey P. Dale, Sep 04 2022 *)

a(n) = A005250(n+1) - 1.

More terms from Warren D. Smith, Dec 11 2000

a(40) corrected by Bert Sierra, Jul 12 2025

A008950 Increasing length runs of consecutive composite numbers (starting points).

Original entry on oeis.org

4, 8, 24, 90, 114, 524, 888, 1130, 1328, 9552, 15684, 19610, 31398, 155922, 360654, 370262, 492114, 1349534, 1357202, 2010734, 4652354, 17051708, 20831324, 47326694, 122164748, 189695660, 191912784, 387096134, 436273010, 1294268492

Offset: 1

Mark Cramer (m.cramer(AT)qut.edu.au). Computed by Dennis Yelle (dennis(AT)netcom.com)

There are runs of n consecutive composite numbers for every n. For example, the n numbers (n+1)!+2 ... (n+1)!+n+1 are composite. Such a run may start of course earlier than this. - Joerg Arndt, May 01 2013

Jens Kruse Andersen, Table of n, a(n) for n=1..74 (using A002386)
Jens Kruse Andersen, The Top-20 Prime Gaps
Jens Kruse Andersen, New record prime gap
Jens Kruse Andersen, Maximal gaps
Eric Weisstein's World of Mathematics, Prime Gaps.
J. Young and A. Potler, First occurrence prime gaps, Math. Comp., 52 (1989), 221-224.

Cf. A008995, A008996.

maxGap = 1; Reap[Do[p = Prime[n]; gap = Prime[n+1] - p; If[gap > maxGap, Print[p+1]; Sow[p+1]; maxGap = gap], {n, 2, 10^8 }]][[2, 1]] (* Jean-François Alcover, Jun 15 2012 *)

a(n) = A002386(n+1)+1.

a(n) <= (n+1)! + 2. [Joerg Arndt, May 01 2013]

A030296 Smallest start for a run of at least n composite numbers.

Original entry on oeis.org

4, 8, 8, 24, 24, 90, 90, 114, 114, 114, 114, 114, 114, 524, 524, 524, 524, 888, 888, 1130, 1130, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 1328, 9552, 9552, 15684, 15684, 15684, 15684, 15684, 15684, 15684, 15684, 19610, 19610, 19610

Offset: 1

Eric W. Weisstein

a(n) is even, since a(n)-1 is a prime > 2, by the minimality of a(n). - Jonathan Sondow, May 31 2014

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(5) = 24 as 24 is the first of the five consecutive composite numbers 24, 25, 26, 27, 28.

Amarnath Murthy, Some more conjectures on primes and divisors, Smarandache Notions Journal, Vol. 12, No. 1-2-3, Spring 2001.

Donovan Johnson, Table of n, a(n) for n = 1..1475 (terms < 4*10^18)
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Eric Weisstein's World of Mathematics, Prime Gaps

Cf. A008950, A008995, A008996, A000101, A002386, A104138.

a[n_] := a[n] = For[p1 = a[n-1]-1; p2 = NextPrime[p1], True, p1 = p2; p2 = NextPrime[p1], If[ p2-p1-1 >= n, Return[p1+1]]]; a[1] = 4; Table[a[n], {n, 1, 43}] (* Jean-François Alcover, May 24 2012 *)
Module[{nn=20000,cmps},cmps=Table[If[CompositeQ[n],1,0],{n,nn}];Table[ SequencePosition[ cmps,PadRight[{},k,1],1][[1,1]],{k,50}]] (* Harvey P. Dale, Jan 01 2022 *)

a(n) = A104138(n) + 1. - Jonathan Sondow, May 31 2014

A192226 Numbers n such that all integers in the interval (n-2*sqrt(sqrt(n)), n] are composite.

Original entry on oeis.org

1, 28, 36, 96, 120, 121, 122, 123, 124, 125, 126, 146, 147, 148, 189, 190, 207, 208, 209, 210, 219, 220, 221, 222, 249, 250, 292, 302, 303, 304, 305, 306, 326, 327, 328, 329, 330, 346, 477, 478, 519, 520, 533, 534, 535, 536, 537, 538, 539, 540, 630, 672

Offset: 1

Juri-Stepan Gerasimov, Jun 26 2011

nonn

a(14432) = 191913030 is probably the last term. Any further terms must be greater than 1.5 * 10^18. - Charles R Greathouse IV, Jun 30 2011

Charles R Greathouse IV, Table of n, a(n) for n = 1..14432

Cf. A188817, A189025, A008995.

Select[Range[700],NextPrime[#-2Sqrt[Sqrt[#]]]>#&] (* Harvey P. Dale, Jul 27 2011 *)

is(n)=for(k=0,sqrtnint(16*n-1,4),if(isprime(n-k), return(0))); 1 \\ Charles R Greathouse IV, Aug 26 2015

a(12)-a(13) added, a(53)-a(56) corrected by Charles R Greathouse IV, Jun 30 2011

A056784 First nonprime in a sequence of consecutive nonprimes which is at least twice as long as any earlier run of consecutive nonprimes in this list.

Original entry on oeis.org

1, 8, 90, 524, 9552, 31398, 2010734, 2300942550, 2614941710600, 352521223451364324

Offset: 1

Frank James Michael Richards, Jun 19 2001

nonn

			2300942550 is the first of 319 consecutive composite numbers.
2614941710600 is the first of 651 consecutive composite numbers.
352521223451364324 is the first of 1327 consecutive composite numbers.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]

Cf. A008950, A008995, A008996, A114100, A002386.

Corrected by R. J. Mathar, Sep 29 2006

Three more terms from Donovan Johnson, Jan 28 2008

cp's OEIS Frontend

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

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A008996 Increasing length runs of consecutive composite numbers (records).

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A008950 Increasing length runs of consecutive composite numbers (starting points).

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A030296 Smallest start for a run of at least n composite numbers.

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A192226 Numbers n such that all integers in the interval (n-2*sqrt(sqrt(n)), n] are composite.

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A056784 First nonprime in a sequence of consecutive nonprimes which is at least twice as long as any earlier run of consecutive nonprimes in this list.

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