A113274 -id:A113274 - 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

A261701 Initial member of four twin prime pairs with gap 210 between them.

Original entry on oeis.org

599, 3917, 5021, 37361, 48779, 81929, 93281, 97157, 263399, 433049, 783149, 821801, 906119, 908669, 1197197, 1308497, 1308707, 1379237, 1464809, 1908449, 2036861, 2341979, 2408561, 2760671, 2804309, 3042491, 3042701, 3042911, 3198197, 4090649, 4543991, 5543927

Offset: 1

K. D. Bajpai, Aug 28 2015

nonn

More precisely, primes p such that p + 2, p + 210, p + 212, p + 420, p + 422, p + 630, p + 632 are all primes.

All the terms in this sequence are congruent to 2 (mod 3).

			599 appears in the sequence because: (a) {599,601}, {809, 811}, {1019, 1021}, {1229, 1231} are four (not consecutive) twin prime pairs; (b) the gap between each twin prime pair (809 - 599) = (1019 -  809) = (1229 - 1019) = 210.

K. D. Bajpai and Dana Jacobsen, Table of n, a(n) for n = 1..10000 [first 865 terms from K. D. Bajpai]
Luis Rodriguez and Carlos Rivera, Gaps between consecutive twin pairs, The Prime Puzzles and Problems Connection.

Cf. A001359 (twin primes), A077800, A113274, A253624.

[p: p in PrimesUpTo (100000) | IsPrime(p+2) and IsPrime(p+210) and IsPrime(p+212) and IsPrime(p+420) and IsPrime(p+422) and IsPrime(p+630) and IsPrime(p+632) ];

select(p -> andmap(isprime, [p, p+2, p+210, p+212, p+420, p+422, p+630, p+632]),[seq(p, p=1..10^5)]);

k = 210; Select[Prime@Range[10^7], PrimeQ[# + 2] && PrimeQ[# + k] && PrimeQ[# + k + 2] && PrimeQ[# + 2 k] && PrimeQ[# + 2 k + 2] && PrimeQ[# + 3 k] &&  PrimeQ[# + 3 k + 2] &]
Select[Prime[Range[400000]],AllTrue[#+{2,210,212,420,422,630,632},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 17 2019 *)

forprime(p= 1, 100000, if(isprime(p+2) && isprime(p+210) && isprime(p+212) && isprime(p+420) && isprime(p+422) && isprime(p+630) && isprime(p+632), print1(p,", ")));

use ntheory ":all"; say join ", ", grep { is_prime($+210) && is_prime($+212) && is_prime($+420) && is_prime($+422) && is_prime($+630) && is_prime($+632) } @{twin_primes(1e8)}; # Dana Jacobsen, Sep 02 2015

use ntheory ":all"; say for sieve_prime_cluster(1, 1e8, 2, 210, 212, 420, 422, 630, 632); # Dana Jacobsen, Oct 03 2015

A261731 Initial member of five twin prime pairs with gap 210 between them.

Original entry on oeis.org

1308497, 3042491, 3042701, 7445309, 20031101, 31572521, 44687987, 54266291, 141208619, 182316521, 237416369, 357080021, 448436321, 611641187, 699458411, 761126027, 774997367, 794065967, 836452961, 915215591, 944958941, 1009194617, 1581935939, 1763255561, 1871007371

Offset: 1

K. D. Bajpai, Aug 30 2015

nonn

More precisely, primes p such that p+2, p+210, p+212, p+420, p+422, p+630, p+632, p+840, p+842 are all primes.

All the terms in this sequence are congruent to 2 (mod 3).

			1308497 appears in this sequence because: (a) {1308497, 1308499}, {1308707, 1308709}, {1308917, 1308919}, {1309127, 1309129}, and {1309337, 1309339} are five twin prime pairs; (b) the gap between each twin prime pair {1308707 - 1308497} = {1308917-1308707} = {1309127 - 1308917} = {1309337 - 1309127} = 210.

K. D. Bajpai and Dana Jacobsen, Table of n, a(n) for n = 1..10000 [first 46 terms from K. D. Bajpai]

Cf. A077800, A113274, A253624, A261701.

[p: p in PrimesUpTo (100000) | IsPrime(p+2) and IsPrime(p+210) and IsPrime(p+212) and IsPrime(p+420) and IsPrime(p+422) and IsPrime(p+630) and IsPrime(p+632) and IsPrime(p+840) and IsPrime(p+842) ];

select(p -> andmap(isprime, [p, p+2, p+210, p+212, p+420, p+422, p+630, p+632, p+840, p+842]),[seq(p, p=1..2*10^7)]);

k = 210; Select[Prime@Range[6*10^7], PrimeQ[# + 2] && PrimeQ[# + k] && PrimeQ[# + k + 2] && PrimeQ[# + 2 k] && PrimeQ[# + 2 k + 2] && PrimeQ[# + 3 k] &&   PrimeQ[# + 3 k + 2] && PrimeQ[# + 4 k] && PrimeQ[# + 4 k + 2] &]
Select[Prime[Range[93*10^6]],AllTrue[#+{2,210,212,420,422,630,632,840,842},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 05 2018 *)

forprime(p= 1,3*10^9, if(isprime(p+2) && isprime(p+210) && isprime(p+212) && isprime(p+420) && isprime(p+422) && isprime(p+630) && isprime(p+632) && isprime(p+840) && isprime(p+842), print1(p,", ")));

use ntheory ":all"; say for sieve_prime_cluster(1,1e10, 2, 210, 212, 420, 422, 630, 632, 840, 842); # Dana Jacobsen, Oct 02 2015

A340573 a(n) is the smallest lesser twin prime p from A001359 such that the distance to the previous lesser twin prime is 6*n.

Original entry on oeis.org

11, 29, 59, 641, 101, 347, 2309, 569, 1931, 521, 1787, 419, 1229, 1871, 3671, 2237, 6551, 1427, 21491, 1607, 12377, 4931, 1019, 23201, 809, 19697, 12539, 2549, 38921, 10709, 37547, 8819, 9239, 34031, 6089, 80447, 15581, 46049, 36341, 14867, 38237, 36779, 87509, 71261, 15137, 40427, 13679, 54917, 41141, 50891

Offset: 1

Artur Jasinski, Jan 12 2021

nonn

Lesser twin primes (with the exception of prime 3) are congruent to 5 modulo 6, which implies that distances between successive pairs of twin primes are 6*k.

			a(1)=11 because 11 - 5 = 6*1.
a(2)=41 because 41 - 29 = 6*2.
a(3)=59 because 59 - 41 = 6*3.

Martin Raab, Table of n, a(n) for n = 1..4508

Cf. A001359, A001359, A053319, A007530, A052380, A052381, A113274, A113275, A052350.

Table[a[n] = 0, {n, 1, 10000}]; Table[
b[n] = 0, {n, 1, 10000}]; qq = {}; prev = 5; Do[
If[Prime[n + 1] - Prime[n] == 2, k = (Prime[n] - prev)/6;
  If[b[k] == 0, a[k] = Prime[n]; b[k] = 1]; prev = Prime[n]], {n, 5,
  10000}]; list = Table[a[n], {n, 1, 50}]
(* Second program: *)
pp = Select[Prime[Range[10^4]], PrimeQ[#+2]&];
dd = Differences[pp];
a[n_] := pp[[FirstPosition[dd, 6n][[1]]+1]];
Array[a, 50] (* Jean-François Alcover, Jan 13 2021 *)

a(n) = A052350(n) + 6*n.

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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A261701 Initial member of four twin prime pairs with gap 210 between them.

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A261731 Initial member of five twin prime pairs with gap 210 between them.

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A340573 a(n) is the smallest lesser twin prime p from A001359 such that the distance to the previous lesser twin prime is 6*n.

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