A002386 -id:A002386 - cp's OEIS frontend

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

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

A036062 Increasing gaps among twin primes: the smallest prime of the second twin pair.

Original entry on oeis.org

5, 11, 29, 59, 101, 347, 419, 809, 2549, 6089, 13679, 18911, 24917, 62927, 188831, 688451, 689459, 851801, 2870471, 4871441, 9925709, 14658419, 17384669, 30754487, 32825201, 96896909, 136286441, 234970031, 248644217, 255953429

Offset: 1

N. J. A. Sloane

nonn

Martin Raab, Table of n, a(n) for n = 1..82 (terms up to a(75) from Max Alekseyev)
Randall Rathbun, Twin Prime Gaps, NMBRTHRY Mailing List, Nov 23 1998.
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.
Tomás Oliveira e Silva, Gaps between twin primes
Eric Weisstein's World of Mathematics, Prime Constellation

Cf. A036061, A036063, A002386, A113274, A113275.

a(n) = A036061(n) + A036063(n).

Terms a(3)-a(41) are given by Rathbun (1998).
Corrected by Jud McCranie, Jan 04 2001
Terms up to a(72) are listed in Kourbatov (2013), terms up to a(75) on Oliveira e Silva's website, added by Max Alekseyev, Nov 06 2015

A036063 Increasing gaps among twin primes: size.

Original entry on oeis.org

0, 4, 10, 16, 28, 34, 70, 148, 166, 208, 280, 370, 496, 628, 922, 928, 1006, 1450, 1510, 1528, 1720, 1900, 2188, 2254, 2830, 2866, 3010, 3100, 3178, 3478, 3802, 4768, 5290, 6028, 6280, 6472, 6550, 6646, 7048, 7978, 8038, 8992, 9310, 9316, 10198, 10336, 10666, 10708

Offset: 1

N. J. A. Sloane

nonn

Martin Raab, Table of n, a(n) for n = 1..82, terms up to a(75) from Max Alekseyev.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Tomás Oliveira e Silva, Gaps between twin primes
Randall Rathbun, Twin Prime Gaps, NMBRTHRY Mailing List, Nov 23 1998.
Eric Weisstein's World of Mathematics, Prime Constellation

Cf. A036061, A036062, A002386.

a(n) = A036062(n) - A036061(n).

a(n) = A113274(n)-2.

Terms 0, 4 prepended, missing term 1006 inserted, and more terms added from A113274 by Max Alekseyev, Nov 05 2015

A085237 Nondecreasing gaps between primes.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 8, 14, 14, 14, 18, 20, 22, 34, 34, 36, 36, 36, 44, 52, 52, 72, 86, 86, 96, 112, 114, 118, 132, 132, 148, 154, 154, 154, 180, 210, 220, 222, 234, 248, 250, 250, 282, 288, 292, 320, 336, 336, 354, 382, 384, 394, 456, 464, 468, 474, 486, 490, 500, 514, 516, 532, 534, 540, 582, 588, 602, 652, 674, 716, 766, 778

Offset: 1

Farideh Firoozbakht, Aug 11 2003

nonn

All terms of A005250 are in the sequence, but some terms of A005250 appear in this sequence more than once.

a(n) is the gap between the n-th and (n+1)-th sublists of prime numbers defined in A348178. - Ya-Ping Lu, Oct 19 2021

			a(21) = a(22) = 34 because prime(218) - prime(217) = prime(1060) - prime(1059) = 34 and prime(n+1) - prime(n) is less than 34, for n < 1059 and n not equal to 217.

R. K. Guy, Unsolved problems in number theory.

Cf. A005250, A005669, A053686, A002386, A348178.

f[n_] := Prime[n+1]-Prime[n]; v={}; Do[ If[f[n]>=If[n==1, 1, v[[ -1]]], v1=n; v=Append[v, f[v1]]; Print[v]], {n, 105000000}]
DeleteDuplicates[Differences[Prime[Range[10^7]]],Greater] (* Harvey P. Dale, Jan 17 2024 *)

from sympy import nextprime; p, r = 2, 0
while r < 778:
    q = nextprime(p); g = q - p
    if g >= r: print(g, end = ', '); r = g
    p = q # Ya-Ping Lu, Jan 23 2024

a(53)-a(63) from Donovan Johnson, Nov 24 2008
a(64)-a(76) from Charles R Greathouse IV, May 09 2011
a(77)-a(79) from Charles R Greathouse IV, May 19 2011

A214757 a(n) = smallest Ramanujan prime R_k in A104272 that is >= A000101(n).

Original entry on oeis.org

11, 11, 11, 29, 97, 127, 569, 937, 1151, 1367, 9613, 15727, 19681, 31481, 156007, 360769, 370387, 492251, 1349669, 1357333, 2010881, 4652507, 17051981, 20831639, 47326913, 122165059, 189695893, 191913047

Offset: 1

John W. Nicholson, Jul 27 2012

nonn

While many values in a(n) are equal to A000101(n), for A214756 it seems the only value such that A002386(n) is equal to A214756(n) is A214756(1) = R_k = A002386(1) = 2.

See "Let rho(m) = A179196(m)" comment at A001223.

			A104272(95) = R_k = 1367 > 1361 = A000101(10), so a(10) = 1367.

Cf. A000101, A104272, A214756, A001223.

Cf. A002386, A214924, A214925, A214926, A214757.

Edited by N. J. A. Sloane, Aug 06 2012

a(16)-a(28) from Donovan Johnson, Nov 04 2012

A349995 Record gaps between odd squarefree semiprimes (A046388).

Original entry on oeis.org

6, 12, 16, 20, 22, 24, 26, 28, 32, 36, 38, 40, 44, 50, 52, 60, 64, 70, 74, 84, 90, 92, 100, 102, 116, 118, 120, 132, 136, 138, 140, 142, 146, 152, 154, 156, 164, 170, 184, 186, 210

Offset: 1

Hugo Pfoertner, Dec 25 2021

			  n  A350098(n)  A350099(n)  a(n)
  1      15          21        6
  2      21          33       12
  3      95         111       16
  4     267         287       20
  5    2369        2391       22

Lucas A. Brown, Python program.

Records in A341828.
Cf. A350098 lower ends of the record gaps, A350099 upper ends of the record gaps.
Cf. A000101, A002386, A005250, A085809, A114057.

a(35)-a(41) from Lucas A. Brown, Feb 29 2024

A053303 Length of maximal prime gap p_{k+1} - p_k with starting prime p_k < 10^n.

Original entry on oeis.org

4, 8, 20, 36, 72, 114, 154, 220, 282, 354, 464, 540, 674, 804, 906, 1132

Offset: 1

Enoch Haga, Mar 05 2000

nonn

Prime gaps associated with A053302.

a(17) is probably 1220 and a(19) is probably 1296. - Robert G. Wilson v, Mar 16 2004

			a(1) = 4 from 7 to 11. a(2) = 8 from 89 to 97. a(3) = 20 from 887 to 907.
a(5)=72 because the 5-digit prime 31397 begins a gap of 72.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Eric Weisstein's World of Mathematics, Prime Gaps

p_k's are in A053302. Cf. A005250, A002386. Essentially the same as A038460.

a(16) from Eric W. Weisstein, Mar 05 2004

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

A073861 Smaller of pair of successive n-digit primes with maximal difference.

Original entry on oeis.org

3, 89, 887, 9551, 31397, 492113, 4652353, 47326693, 436273009, 4302407359, 42652618343, 738832927927, 7177162611713, 90874329411493, 218209405436543, 1693182318746371, 80873624627234849, 804212830686677669

Offset: 1

Amarnath Murthy, Aug 15 2002

A subsequence of A002386 (assuming that there is an n-digit term in A002386 for all values of n). - M. F. Hasler, Apr 28 2014

			a(3) = 887, the next prime is 907, 907-887=20 is the maximal possible difference of two 3-digit primes and no smaller pair exhibits this property.

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

Table[Last[Sort[{#[[2]]-#[[1]],#[[1]],#[[2]]}&/@Partition[Prime[Range[PrimePi[10^i]+1,PrimePi[10^(i+1)]]],2,1]]][[2]],{i,7}] (* Harvey P. Dale, Jan 23 2010 *)

A073861 = A000040 o A241623. - M. F. Hasler, Apr 28 2014

a(n) = max { p in A002386 | nextprime(p) < 10^n } (under the assumption given in the comment). - M. F. Hasler, Apr 28 2014

Corrected error and added terms Harvey P. Dale, Jan 23 2010

a(9)-a(18) from Donovan Johnson, Nov 29 2010

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

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

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A036062 Increasing gaps among twin primes: the smallest prime of the second twin pair.

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A036063 Increasing gaps among twin primes: size.

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A085237 Nondecreasing gaps between primes.

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A214757 a(n) = smallest Ramanujan prime R_k in A104272 that is >= A000101(n).

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A349995 Record gaps between odd squarefree semiprimes (A046388).

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A053303 Length of maximal prime gap p_{k+1} - p_k with starting prime p_k < 10^n.

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

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A073861 Smaller of pair of successive n-digit primes with maximal difference.

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