A008950 -id:A008950 - 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

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

A008995 Increasing length runs of consecutive composite numbers (endpoints).

Original entry on oeis.org

4, 10, 28, 96, 126, 540, 906, 1150, 1360, 9586, 15726, 19660, 31468, 156006, 360748, 370372, 492226, 1349650, 1357332, 2010880, 4652506, 17051886, 20831532, 47326912, 122164968, 189695892, 191913030

Offset: 1

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

nonn

Netnews group rec.puzzles, circa Mar 01 1996 (I would like to get the exact reference).

MathForum rec.puzzles archive, Arithmetic Question 8 - consecutive.composites, June 2005.
User "Abigail", 1000 consectutive composites (sic), post in newsgroup rec.puzzles, Jun 19 1996.
Eric Weisstein's World of Mathematics, Prime Gaps.

Cf. A000101, A008950, A008996.

maxGap = 1; Reap[ Do[ gap = Prime[n + 1] - (p = Prime[n]); If[gap > maxGap, Print[p + gap - 1]; Sow[p + gap - 1]; maxGap = gap], {n, 2, 10^8}]][[2, 1]] (* Jean-François Alcover, Jun 12 2013 *)

a(n) = A000101(n+1)-1.

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

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

A383969 a(n) is the smallest even number m such that the set {m+1, m+3, m+5, ..., m+(2*n-1)} contains no prime numbers.

Original entry on oeis.org

0, 24, 90, 114, 114, 114, 524, 524, 888, 1130, 1328, 1328, 1328, 1328, 1328, 1328, 9552, 15684, 15684, 15684, 15684, 19610, 19610, 19610, 19610, 31398, 31398, 31398, 31398, 31398, 31398, 31398, 31398, 31398, 31398, 155922, 155922, 155922, 155922, 155922, 155922, 155922

Offset: 1

David James Sycamore, May 16 2025

nonn

For a(2) onward, consists of terms of A008950(n), perhaps repeated, based on the increase in records. So, higher terms may be easily computed using the b-files at A008950 and A002386. - Michael S. Branicky, May 23 2025

			a(1) = 0 since 0 is the smallest even number such that 0 + 1 = 1 is not prime
a(2) = 24 since there are no primes in {24+1, 24+3} = {25, 27} and no smaller even number has this property.
a(3) = 90 since there are no primes in {91, 93, 95}, and no smaller even number has this property.

Cf. A000040, A002808, A018252, A008950, A002386.

k = 0; Table[While[AnyTrue[k + Range[2*n - 1], PrimeQ], k += 2]; k, {n, 42}] (* Michael De Vlieger, Jun 01 2025 *)

from sympy import isprime
from itertools import count
def a(n): return next(m for m in count(0, 2) if all(not isprime(m+2*i+1) for i in range(n)))
print([a(n) for n in range(1, 43)]) # Michael S. Branicky, May 23 2025

a(4) and beyond from Michael S. Branicky, May 23 2025

A175839 Smallest runs of n*2-1 consecutive composites.

Original entry on oeis.org

4, 8, 9, 10, 24, 25, 26, 27, 28, 90, 91, 92, 93, 94, 95, 96, 114, 115, 116, 117, 118, 119, 120, 121, 122, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126

Offset: 1

Grant Garcia, Sep 20 2010

Even lengths of runs of composites are omitted, as nontrivial runs always have odd lengths (see A046933).

Run 5 (starting at 114) has 13 consecutive composites and is repeated in runs 6 and 7. Run 8 starts at 524 = A008950(6).

			Run 1 has length 1; the first composite is 4.
Run 2 has length 3; the first three consecutive composites are 8, 9, and 10.
Run 3 has length 5; the first five consecutive composites are 24, 25, 26, 27, and 28.
4;
8, 9, 10;
24, 25, 26, 27, 28;
90, 91, 92, 93, 94, 95, 96;
114, 115, 116, 117, 118, 119, 120, 121, 122;
114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124;
114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126;

Wikipedia, Composite number

Cf. A002808, A046933, A008950.

from sympy import isprime
out = []
run = 1
for n in range(4, 10000):
    isrun = True
    for o in range(run): isrun *= not isprime(n + o - run)
    if isrun:
        for o in range(run): out.append(n + o - run)
        run += 2
print(out)

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

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A008995 Increasing length runs of consecutive composite numbers (endpoints).

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

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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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A383969 a(n) is the smallest even number m such that the set {m+1, m+3, m+5, ..., m+(2*n-1)} contains no prime numbers.

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A175839 Smallest runs of n*2-1 consecutive composites.

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