A008996 -id:A008996 - cp's OEIS frontend

A046933 Number of composites between successive primes.

0, 1, 1, 3, 1, 3, 1, 3, 5, 1, 5, 3, 1, 3, 5, 5, 1, 5, 3, 1, 5, 3, 5, 7, 3, 1, 3, 1, 3, 13, 3, 5, 1, 9, 1, 5, 5, 3, 5, 5, 1, 9, 1, 3, 1, 11, 11, 3, 1, 3, 5, 1, 9, 5, 5, 5, 1, 5, 3, 1, 9, 13, 3, 1, 3, 13, 5, 9, 1, 3, 5, 7, 5, 5, 3, 5, 7, 3, 7, 9, 1, 9, 1, 5, 3, 5, 7, 3, 1, 3, 11, 7, 3, 7, 3, 5, 11, 1, 17

Offset: 1

Marc LeBrun, Dec 11 1999

a(n) is odd for n>1 since all primes except 2 are odd. - Joel Brennan, Jan 02 2023

			a(1) = 0 since 2 is adjacent to 3;
a(2) = 1 since 4 is between 3 and 5;
a(4) = 3 = 11 - 7 - 1, etc.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000040, A028334, A036263.

Cf. A008996 (record values > 0).

a046933 n = a046933_list !! (n-1)
a046933_list = map (subtract 1) a001223_list
-- Reinhard Zumkeller, Dec 12 2012

A046933:=n->ithprime(n+1)-ithprime(n)-1; seq(A046933(n), n=1..100); # Wesley Ivan Hurt, Apr 15 2014

Differences[Prime[Range[100]]] - 1 (* Arkadiusz Wesolowski, Nov 18 2011 *)
Table[Prime[n + 1] - Prime[n] - 1, {n, 100}] (* Wesley Ivan Hurt, Apr 15 2014 *)
Prepend[Drop[Length/@SequenceSplit[Range@Prime@100,{?PrimeQ}],1],0] (* _Federico Provvedi, Jul 19 2021 *)

a(n)=prime(n+1)-prime(n)-1 \\ Charles R Greathouse IV, Nov 20 2012

from sympy import prime
def A046933(n): return prime(n+1)-prime(n)-1 # Chai Wah Wu, Jan 02 2024

a(n) = prime(n+1) - prime(n) - 1 = A000040(n+1) - A000040(n) - 1.
a(n) = A001223(n) - 1.
a(n) = 2*A028334(n) - 1 for n>1. - Giovanni Teofilatto, Apr 19 2010
a(n) = Sum_{i=1..n-1} A036263(i). - Daniel Forgues, Apr 07 2014

A005250 Record gaps between primes.

Original entry on oeis.org

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

A058320 Distinct even prime-gap lengths (number of composites between primes), from 3+2, 7+4, 23+6,...

Original entry on oeis.org

2, 4, 6, 8, 14, 10, 12, 18, 20, 22, 34, 24, 16, 26, 28, 30, 32, 36, 44, 42, 40, 52, 48, 38, 72, 50, 62, 54, 60, 58, 46, 56, 64, 68, 86, 66, 70, 78, 76, 82, 96, 112, 100, 74, 90, 84, 114, 80, 88, 98, 92, 106, 94, 118, 132, 104, 102, 110, 126, 120, 148, 108

Offset: 0

Warren D. Smith, Dec 11 2000

Nicely and Nyman have sieved up to 1.3565*10^16 at least. They admit it is likely they have suffered from hardware or software bugs, but believe the probability the sequence up to this point is incorrect is <1 in a million. This sequence is presumably all even integers (in different order). It is not monotonic. The monotonic subsequence of record-breaking prime gaps is A005250.

Essentially the same as A014320. [From R. J. Mathar, Oct 13 2008]

Richard P. Brent, The first occurrence of large gaps between successive primes, Math. Comp. 27:124 (1973), 959-963.
Thomas R. Nicely, New maximal prime gaps and first occurrences, Math. Comput. 68,227 (1999) 1311-1315.
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]

Cf. A008996, A005250.

Equals 2*A014321(n-1).

DeleteDuplicates[Differences[Prime[Range[2,200000]]]] (* Harvey P. Dale, Dec 07 2014 *)

Comment corrected by Harvey P. Dale, Dec 07 2014

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]

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

A084105 Middle q of three consecutive primes p,q,r, such that one adjacent prime is near, the other is far and the ratio of the differences (whichever of (r-q)/(q-p) or (q-p)/(r-q) is greater than 1) sets a record.

Original entry on oeis.org

3, 29, 113, 139, 199, 523, 1151, 1669, 2971, 6947, 10007, 16141, 25471, 40639, 79699, 102761, 173359, 265621, 404851, 838249, 1349533, 1562051, 6371537, 7230479, 27980987, 42082303, 53231051, 70396589, 192983851, 253878617, 390932389, 465828731, 516540163, 1692327137

Offset: 1

Hugo Pfoertner, May 29 2003

nonn

Are there entries other than a(3) for which the smaller difference exceeds 2?

			a(3) = 113 because the ratio (113-109)/(127-113) = 2/7 = 0.28571.. is smaller than the previous minimum produced by (31-29)/(29-23) = 1/3 = 0.33333...

Martin Ehrenstein, Table of n, a(n) for n = 1..57
Hugo Pfoertner, Maximally asymmetric prime triples, FORTRAN program

Cf. A084106, A031132, A008996, A002386, A329158, A329159, A329160, A329161, A337489.

a084105(limit)={my(p1=2,p2=3,r=0);forprime(p3=5,limit,my(q=max((p2-p1)/(p3-p2),(p3-p2)/(p2-p1)));if(q>r,r=q;print1(p2,", "));p1=p2;p2=p3)};
a084105(600000000) \\ Hugo Pfoertner, Sep 04 2020

More terms from Don Reble, May 29 2003

a(32)-a(34) from Hugo Pfoertner, Nov 06 2019

cp's OEIS Frontend

A046933 Number of composites between successive primes.

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A005250 Record gaps between primes.

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A058320 Distinct even prime-gap lengths (number of composites between primes), from 3+2, 7+4, 23+6,...

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

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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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A084105 Middle q of three consecutive primes p,q,r, such that one adjacent prime is near, the other is far and the ratio of the differences (whichever of (r-q)/(q-p) or (q-p)/(r-q) is greater than 1) sets a record.

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