A036061 -id:A036061 - cp's OEIS frontend

A091592 Numbers n such that there are no twin primes between n^2 and (n+1)^2.

1, 9, 19, 26, 27, 30, 34, 39, 49, 53, 77, 122

Offset: 1

Hugo Pfoertner, Jan 25 2004

Numbers n such that there is no pair of twin primes P, P+2 with n^2 < P < P+2 < n^2+2*n.
The first 7 terms of this sequence were given by Ernst Jung in a discussion in the Newsgroup de.sci.mathematik entitled "Primzahlen zwischen (2x-1)^2 und (2x+1)^2" (primes between ...and...) with other significant contributions from Hermann Kremer and Rainer Rosenthal. It is conjectured that there are no further terms beyond a(12)=122. This has been tested to 50000 by Robert G. Wilson v.
Tested up to 10^7 and found no such numbers. - Arkadiusz Wesolowski, Jul 11 2011

			9 is a term because no twin primes are found in the interval [9^2,10^2].

J. Korevaar, The prime-pair conjectures of Hardy and Littlewood, Indagationes Mathematicae, Volume 23, Issue 3, 2012, Pages 269-299.
A. Kourbatov, Maximal Gaps Between Prime k-Tuples: A Statistical Approach, J. Int. Seq. 16 (2013) #13.5.2
Hugo Pfoertner, Illustration of record gaps between pairs of twin primes.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.
Eric Weisstein's World of Mathematics, Twin Prime Conjecture.

Cf. A091591, A036061, A036063, A113274.

isA091592 := proc(n) local p; p := nextprime(n^2) ; q := nextprime(p) ; while q < n^2+2*n do if q-p = 2 then RETURN(false) ; fi; p :=q ; q := nextprime(p) ; od: RETURN(true) ; end: for n from 1 do if isA091592(n) then printf("%d ",n) ; fi; od: # R. J. Mathar, Aug 26 2008

fQ[n_] := StringCount[ ToString@ PrimeQ[ Range[n^2, (n + 1)^2]], "True, False, True"] == 0; lst = {}; Do[ If[ fQ@n, AppendTo[lst, n]], {n, 25000}]

Edited by N. J. A. Sloane, Aug 31 2008 at the suggestion of Pierre CAMI

A113275 Lesser of twin primes for which the gap before the following twin primes is a record.

Original entry on oeis.org

3, 5, 17, 41, 71, 311, 347, 659, 2381, 5879, 13397, 18539, 24419, 62297, 187907, 687521, 688451, 850349, 2868959, 4869911, 9923987, 14656517, 17382479, 30752231, 32822369, 96894041, 136283429, 234966929, 248641037, 255949949

Offset: 1

Bernardo Boncompagni, Oct 21 2005

nonn

			The smallest twin prime pair is 3, 5, then 5, 7 so a(1) = 3; the following pair is 11, 13 so a(2) = 5 because 11 - 5 = 6 > 5 - 3 = 2; the following pair is 17, 19: since 17 - 11 = 6 = 11 - 5 nothing happens; the following pair is 29, 31 so a(3)= 17 because 29 - 17 = 12 > 11 - 5 = 6.

Martin Raab, Table of n, a(n) for n = 1..82 (first 75 terms from Max Alekseyev)
Harvey Dubner, Twin Prime Conjectures, Journal of Recreational Mathematics, Vol. 30 (3), 1999-2000.
Alexei Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013.
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.
Mersenneforum, Gaps between prime pairs (Twin Primes).
Tomás Oliveira e Silva, Gaps between twin primes

Record gaps are given in A113274. Cf. A002386.

NextLowerTwinPrim[n_] := Block[{k = n + 2}, While[ !PrimeQ[k] || !PrimeQ[k + 2], k++ ]; k]; p = 3; r = 0; t = {3}; Do[q = NextLowerTwinPrim[p]; If[q > r + p, AppendTo[t, p]; r = q - p]; p = q, {n, 10^9}] (* Robert G. Wilson v, Oct 22 2005 *)

a(n) = A036061(n) - 2.

a(n) = A036062(n) - A113274(n).

a(22)-a(30) from Robert G. Wilson v, Oct 22 2005

Terms up to a(72) are listed in Kourbatov (2013), terms up to a(75) in Oliveira e Silva's website, added by Max Alekseyev, Nov 06 2015

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

A054691 New records in A054690 (start of n consecutive non-twin primes).

Original entry on oeis.org

7, 19, 43, 73, 349, 661, 8629, 13399, 14629, 24421, 62299, 187909, 688453, 850351, 17382481, 30752233, 32822371, 136283431, 248641039, 255949951, 390817729, 698542489, 1256735191, 1535220499, 1899797989, 2466641071, 4289385523, 24215097499, 42441715489, 43725662623

Offset: 1

Jeff Burch, Apr 19 2000

nonn

Has many terms in common with A036061, but neither of the two is a subsequence of the other one. - M. F. Hasler, May 07 2022

Amiram Eldar, Table of n, a(n) for n = 1..41

Cf. A054690, A054692, A242459.

Cf. A036061.

from sympy import nextprime
from itertools import count, islice
def agen(): # generator of terms
    p, q, n, rl, argp = 2, 3, 1, 0, 2
    for k in count(1):
        if q - p >= 4: rl += 1
        else:
            if rl >= n: yield argp; n = rl + 1
            rl, argp = 0, q
        p, q = q, nextprime(q)
print(list(islice(agen(), 14))) # Michael S. Branicky, Aug 23 2022

a(10)-a(25) from Sean A. Irvine, Feb 17 2022

Offset changed and a(26)-a(29) from Michael S. Branicky, Aug 23 2022

cp's OEIS Frontend

A091592 Numbers n such that there are no twin primes between n^2 and (n+1)^2.

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A113275 Lesser of twin primes for which the gap before the following twin primes is a record.

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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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A054691 New records in A054690 (start of n consecutive non-twin primes).

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