A087641 -id:A087641 - cp's OEIS frontend

A035789 Start of a string of exactly 1 consecutive (but disjoint) pair of twin primes.

29, 41, 59, 71, 227, 239, 269, 281, 311, 347, 461, 521, 569, 599, 617, 641, 659, 857, 881, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1607, 1619, 1667, 1697, 1721, 1787, 1997, 2027, 2141, 2237, 2267, 2309, 2339, 2381, 2549, 2591, 2657, 2687

Offset: 1

Randall L Rathbun, Nov 30 1998

Lesser of lonely twin primes.

Old Name was: Let P1,P2,..,P6 be any 6 consecutive primes. The sequence consists of those values of P3 for which P2-P1>2, P4-P3=2 and P6-P5>2.

			The first lonely twin primes (A069453) are 29,31 (23 and 37 are non-twin), 41,43 (37 and 47 are non-twin), 59,61 (53 and 67 are non-twin). Of these, the lesser twins are 29,41,59, so this is how the sequence begins.
23, 27, 29, 31, 37, 41: 27-23>2, 31-29=2, 41-37>2; so 29 is in the sequence.
From _Hartmut F. W. Hoft_, Apr 05 2016: (Start)
The example should read: 19, 23, 29, 31, 37, 41: 23-19>2, 31-29=2, 41-37>2; so 29 is in the sequence.
a(n)=A069453(2n-1), n>=1.
(End)

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000
Hugo Pfoertner, FORTRAN program: Consecutive pairs of twin primes.
Randall Rathbun, A study of n-twin_prime clusters among prime numbers, Posting to Number Theory List, Nov 19 1998.

Cf. A035790, A035791, A035792, A035793, A035794, A035795, A087641.

Cf. A069453, A069455.

PrimeNext[n_]:=Module[{k},k=n+1;While[ !PrimeQ[k],k++ ];k]; PrimePrev[n_]:=Module[{k},k=n-1;While[ !PrimeQ[k],k-- ];k]; lst={};Do[p=Prime[n];If[ !PrimeQ[p-2]&&!PrimeQ[p+4]&&PrimeQ[p+2]&&!PrimeQ[PrimePrev[p]-2]&&!PrimeQ[PrimeNext[p+2]+2],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 22 2009 *)
(* starting at n=3 would eliminate the first two primality tests, Hartmut F. W. Hoft, Apr 09 2016 *)

Edited by Hugo Pfoertner, Oct 15 2003

A035790 Start of a string of exactly 2 consecutive (but disjoint) pairs of twin primes.

Original entry on oeis.org

101, 137, 419, 1019, 1049, 1481, 1871, 1931, 2081, 2111, 2969, 3251, 3461, 4259, 5009, 5651, 5867, 6689, 6947, 7331, 7547, 8219, 8969, 10007, 11057, 11159, 11699, 12239, 13001, 13709, 13997, 14561, 15641, 15731, 16061, 16631, 17579, 17909

Offset: 1

Randall L Rathbun, Nov 30 1998

Let P1,P2,..,P8 be any 8 consecutive primes. The sequence consists of those values of P3 for which P2-P1 > 2, P4-P3 = 2, P6-P5= 2 and P8-P7 > 2.

			89, 97, 101, 103, 107, 109, 113, 127: 97-89 > 2, 103-101 = 2, 109-107 = 2, 127-113 > 2.

Posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Nov. 19 1998.

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000
G. L. Honaker, Jr. and Chris Caldwell, 10000...01521 (100-digits)
Hugo Pfoertner, FORTRAN program: Consecutive pairs of twin primes.

Cf. A035789, A035791, A035792, A035793, A035794, A035795, A087641.

Select[Prime@ Range@ 2100, And[NextPrime[#, -1] - NextPrime[#, -2] > 2, NextPrime@ # - # == 2, NextPrime[#, 3] - NextPrime[#, 2] == 2, NextPrime[#, 5] - NextPrime[#, 4] > 2] &] (* Michael De Vlieger, Apr 25 2015 *)

a(n)={L=vector(7);forprime(p=o=1,,L=concat(L[2..7],-o+o=p); L[3]==2&&L[5]==2&&L[1]>2&&L[2]>2&&L[4]>2&&L[6]>2&&L[7]>2&&!n--&&return(p-sum(i=3,7,L[i])))} \\ M. F. Hasler, May 04 2015

a(10)=2111, a(10^2)=77261, a(10^3)=1603697, a(10^4)=27397631, a(10^5)=435140477, a(10^6)=6391490657. - M. F. Hasler, May 04 2015

Edited by Hugo Pfoertner, Oct 15 2003
Offset corrected by Arkadiusz Wesolowski, May 06 2012
Double-checked up to a(10^4)=27397631 by M. F. Hasler, May 04 2015

A035791 Start of a string of exactly 3 consecutive (but disjoint) pairs of twin primes.

Original entry on oeis.org

5, 179, 809, 3359, 4217, 6761, 18041, 21587, 26861, 49367, 67187, 80447, 82721, 91127, 97841, 98897, 103967, 109829, 122597, 154157, 178037, 203321, 208931, 225749, 227609, 236867, 243671, 251201, 266447, 285611, 289109, 295871, 317729

Offset: 1

Randall L Rathbun

			a(2)=179 because (179,181),(191,193),(197,199) is the second occurrence (after (5,7),(11,13),(17,19)) of exactly 3 pairs of twin primes.

Posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Nov. 19 1998.

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000

Cf. A035789, A035790, A035792, A035793, A035794, A035795, A087641.

Select[Prime@ Range@ 30000, And[ NextPrime[#, -1] - NextPrime[#, -2] != 2, NextPrime@ # - # == 2, NextPrime[#, 3] - NextPrime[#, 2] == 2, NextPrime[#, 5] - NextPrime[#, 4] == 2, NextPrime[#, 7] - NextPrime[#, 6] > 2] &] (* Michael De Vlieger, Apr 25 2015 *)
Select[Partition[Prime[Range[30000]],10,1],#[[8]]-#[[7]]==#[[6]]-#[[5]] == #[[4]] - #[[3]]==2&&#[[2]]-#[[1]]!=2&&#[[10]]-#[[9]]!=2&][[All,3]] (* Harvey P. Dale, Mar 14 2018 *)

More terms from Hugo Pfoertner, Sep 05 2003

Offset corrected by Arkadiusz Wesolowski, May 06 2012

A035792 Start of a string of exactly 4 consecutive (but disjoint) pairs of twin primes.

Original entry on oeis.org

9419, 62969, 72221, 392261, 495569, 663569, 1006301, 1138367, 1159187, 1173539, 1322147, 2144477, 2168651, 2502341, 2668217, 3020999, 3215711, 3664679, 4890857, 5248079, 5261699, 5532269, 5561597, 5651729, 5787317, 6256727

Offset: 1

Randall L Rathbun

nonn

Posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Nov. 19 1998.

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000

The first term of this sequence is in A087641.

Cf. A001359, A035789, A035790, A035791, A035792, A035793, A035794, A035795.

Prime[Select[Range[1000000], Prime[ # + 1] - Prime[ # ] == 2 && Prime[ # + 3] - Prime[ # + 2] == 2 && Prime[ # + 5] - Prime[ # + 4] == 2 && Prime[ # + 7] - Prime[ # + 6] == 2 &]] (* Tanya Khovanova, Sep 07 2007 *)

a(11)-a(26) from Hugo Pfoertner, Sep 16 2003

A035795 Start of a string of exactly 7 consecutive (but disjoint) pairs of twin primes.

Original entry on oeis.org

678771479, 17479880399, 17830729991, 23799917819, 70455134039, 79453842029, 108108566471, 150411604619, 163868216387, 256385651969, 444790621787, 446688503687, 496081268777, 502910801927, 688735396829, 711503536589, 712407842477, 793957831409, 808316366171, 881191407827, 891108993767, 896804723201

Offset: 1

Randall L Rathbun

nonn

Tomáš Brada, Table of n, a(n) for n = 1..10000 (terms 23..29 from Vasily Danilov, terms 30..4608 from Dmitry Petukhov)
R. Rathbun, A study of n-twin_prime clusters among primes, NMBRTHRY Mailing List, Nov 19 1998.

The first term of this sequence is in A087641.

More terms from Jud McCranie, Sep 16 2003
Offset corrected by Arkadiusz Wesolowski, May 06 2012
a(17) from Natalia Makarova, Oct 06 2015
a(18)-a(22) from Vasily Danilov, Oct 08 2015

A035793 Start of a string of exactly 5 consecutive (but disjoint) pairs of twin primes.

Original entry on oeis.org

909287, 2596619, 9617981, 12628337, 18873497, 21579629, 25739771, 34140077, 39433367, 62832101, 67369397, 84733211, 90122507, 102243017, 132826607, 140456711, 142749149, 180929687, 201057539, 212461979, 219970547, 228001649

Offset: 1

Randall L Rathbun

nonn

Vasily Danilov, Table of n, a(n) for n = 1..10000 (terms a(23)-a(500) from _Sebastian Petzelberger).
Randall Rathbun, A study of n-twin_prime clusters among prime numbers, Posting to Number Theory List, Nov 19 1998.

Cf. A001359, A035789, A035790, A035791, A035792, A035793, A035794, A035795, A087641.

lst={};Do[p0=Prime[n];p1=Prime[n+1];p2=Prime[n+2];p3=Prime[n+3];p4=Prime[n+4];p5=Prime[n+5];p6=Prime[n+6];p7=Prime[n+7];p8=Prime[n+8];p9=Prime[n+9];If[p1-p0==p3-p2==p5-p4==p7-p6==p9-p8==2,AppendTo[lst,p0]],{n,10!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2010 *)

a(11)-a(22) from Hugo Pfoertner, Sep 16 2003

Offset corrected by Arkadiusz Wesolowski, May 06 2012

A035794 Start of a string of exactly 6 consecutive (but disjoint) pairs of twin primes.

Original entry on oeis.org

325267931, 412984667, 2227604747, 2409360557, 4014288869, 4363839617, 6988064579, 8402566787, 9497780417, 10099096127, 12347083739, 12429980741, 13022601257, 14198015129, 14845029299, 15330685079, 16810761029, 17049454841, 18266059421, 18864084791

Offset: 1

Randall L Rathbun

nonn

Vasily Danilov, Table of n, a(n) for n = 1..10000, a(27)-a(41), a(112), a(227)-a(253) from Natalia Makarova, remaining terms from Vasily Danilov and Dmitry Petukhov.
Randall Rathbun, A study of n-twin_prime clusters among prime numbers, Posting to Number Theory List, Nov 19 1998.

Cf. A001359, A035789, A035790, A035791, A035792, A035793, A035794, A035795, A087641.

fQ[n_] := Block[{k = 6}, And[NextPrime[n, -1] - NextPrime[n, -2] != 2, NextPrime[n, 2 k + 1] - NextPrime[n, 2 k] != 2, AllTrue[NextPrime[n, # + 1] - NextPrime[n, #] & /@ (Range[0, 2 k - 1, 2]), # == 2 &]]]; Select[Prime@ Range[10^9], fQ] (* Michael De Vlieger, May 09 2015, Version 10 *)

isok(p) = {if (! isprime(p-2) && isprime(p+2), for (k=2, 6, my(q = nextprime(p+3)); if (! isprime(q+2), return (0)); p = q+2;); q = nextprime(p+3); if (isprime(q+2), return (0)); return (1);); return (0);} \\ Michel Marcus, Dec 06 2019

a(11)-a(17) from Jud McCranie, Sep 16 2003
Offset corrected by Arkadiusz Wesolowski, May 06 2012
Wrong term 678771479 deleted and a(18)-a(26) from Sebastian Petzelberger, May 04 2015

A348168 Segment the list of prime numbers into sublists L_1, L_2, ... with L_1 = {2} and L_n = {p_1, p_2, ..., p_a(n)}, where a(n) is the largest m such that for 0 < i < m, p_1 - prevprime(p_1) > p_2 - p_1 >= p_{i+1} - p_i.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 2, 4, 1, 2, 3, 2, 1, 6, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 4, 2, 8, 1, 4, 2, 2, 1, 1, 5, 2, 1, 2, 2, 2, 1, 4, 6, 2, 2, 5, 8, 7, 2, 1, 1, 2, 10, 2, 2, 2, 2, 1, 4, 4, 2, 1, 5, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 4, 1, 1, 3, 2, 2, 3, 1, 2, 1, 2, 1, 2

Offset: 1

Ya-Ping Lu, Oct 03 2021

The gap between two consecutive primes in L_n is smaller than g_{n-1} and g_n, where g_n is the gap between L_n and L_{n+1}. Sublists of length 2 are the most frequent ones and any pair of twin primes >= 11 stay in the same sublist.
Conjecture 1: lim_{n->oo} N_i/n = k_i, where N_i is the number of the first n sublists consisting of i primes and k_i is a constant, with k_2 > k_1 > k_3 > k_4 > ... .
Conjecture 2: lim_{n->oo} (Sum_{i=1..n} a(i))/n = Sum_{i=1..oo} i*k_i = e, meaning that, as n tends to infinity, the average length of sublists approaches 2.71828... (see the partial average - n plot in the links).
From Ya-Ping Lu, Apr 15 2024: (Start)
The distribution of sublists with 1, 2, 3, 4 and 5 primes and the number of primes in the first n sublists are given in the table below. k_i's as defined in Conjecture 1 are: k1 = 0.281, k2 = 0.431, k3 = 0.127, k4 = 0.058, and k5 = 0.031, approximately. Sublists with length <= 5 account for about 93% of the terms and 70% of the primes, as n approaches infinity.
 n            N_1        N_2        N_3        N_4       N_5       # of primes
 ----------   ---------  ---------  ---------  --------  --------  -----------
 1            1          0          0          0         0         1
 10           6          3          0          1         0         16
 100          33         44         5          9         3         232
 1000         277        431        120        72        36        2617
 10000        2821       4225       1243       642       331       27214
 100000       28072      42929      12427      6059      3159      276081
 1000000      279751     430299     126008     59729     32043     2747392
 10000000     2804959    4303512    1264532    592726    317127    27426366
 100000000    28070302   43078975   12686566   5869443   3143266   273972452
 1000000000   280903920  431182582  127100032  58293618  31258182  2737643048
(End)

			See also the table of the sublists in the examples for A362017.
a(1) = 1 because L_1 = {2} by definition.
In the following examples we use p_0 to denote prevprime(p_1).
a(2) = 1. For the 2nd sublist, p_1 - p_0 = 3 - 2 = 1. If the next prime, 5, is in L_2, then p_2 - p_1 = 2 > p_1 - p_0. Therefore, 5 does not belong to L_2 and L_2 = {3}.
a(5) = 2. For the 5th sublist, p_1 - p_0 = 11 - 7 = 4. p_2 = 13 is in L_5 because p_2 - p_1 = 2 < p_1 - p_0. However, the next prime, 17, is not in L_5 as 17 - 13 > p_2 - p_1. Thus, L_5 = {11, 13}.
a(15) = 6. L_15 = {97, 101, 103, 107, 109, 113}, because p_1 - p_0 = 97-89 > p_2 - p_1 = 101-97 = 4, which is the maximum prime gap in L_15. 127, the prime after 113, is not in L_15 as 127-113 = 14 > p_2 - p_1.

Cf. A362017 (first in each sublist), A087641, A226657, A001359, A023200.

from sympy import nextprime
L = [2]
for n in range(1, 100):
    print(len(L), end =', ')
    p0 = L[-1]; p1 = nextprime(p0); M = [p1]; g0 = p1 - p0; p = nextprime(p1); g1 = p - p1
    while g1 < g0 and p - p1 <= g1: M.append(p); p1 = p; p = nextprime(p)
    L = M

Edited by Peter Munn, Jul 08 2025

A179067 Orders of consecutive clusters of twin primes.

Original entry on oeis.org

1, 3, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Franz Vrabec, Jun 27 2010

For k>=1, 2k+4 consecutive primes P1, P2, ..., P2k+4 defining a cluster of twin primes of order k iff P2-P1 <> 2, P4-P3 = P6-P5 = ... = P2k+2 - P2k+1 = 2, P2k+4 - P2k+3 <> 2.

Also the lengths of maximal runs of terms differing by 2 in A029707 (leading index of twin primes), complement A049579. - Gus Wiseman, Dec 05 2024

			The twin prime cluster ((101,103),(107,109)) of order k=2 stems from the 2k+4 = 8 consecutive primes (89, 97, 101, 103, 107, 109, 113, 127) because 97-89 <> 2, 103-101 = 109-107 = 2, 127-113 <> 2.
From _Gus Wiseman_, Dec 05 2024: (Start)
The leading indices of twin primes are:
  2, 3, 5, 7, 10, 13, 17, 20, 26, 28, 33, 35, 41, 43, 45, 49, 52, ...
with maximal runs of terms differing by 2:
  {2}, {3,5,7}, {10}, {13}, {17}, {20}, {26,28}, {33,35}, {41,43,45}, {49}, {52}, ...
with lengths a(n).
(End)

Robert Israel, Table of n, a(n) for n = 1..10000
Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

Cf. A077800.
Cf. A001359, A111950, A087641.
Cf. A035789, A035790, A035791, A035792, A035793, A035794, A035795.
A000040 lists the primes, differences A001223 (run-lengths A333254, A373821).
A006512 gives the greater of twin primes.
A029707 gives the leading index of twin primes, complement A049579.
A038664 finds the first prime gap of length 2n.
A046933 counts composite numbers between primes.
A000720, A006560, A006562, A014574, A037201, A107770, A122535, A155752, A175632, A251092.

R:= 1: count:= 1: m:= 0:
q:= 5: state:= 1:
while count < 100 do
 p:= nextprime(q);
 if state = 1 then
    if p-q = 2 then state:= 2; m:= m+1;
    else
      if m > 0 then R:= R,m; count:= count+1; fi;
      m:= 0
    fi
 else state:= 1;
 fi;
 q:= p
od:
R; # Robert Israel, Feb 07 2023

Length/@Split[Select[Range[2,100],Prime[#+1]-Prime[#]==2&],#2==#1+2&] (* Gus Wiseman, Dec 05 2024 *)

a(n)={my(o,P,L=vector(3));n++;forprime(p=o=3,,L=concat(L[2..3],-o+o=p);L[3]==2||next;L[1]==2&&(P=concat(P,p))&&next;n--||return(#P);P=[p])} \\ M. F. Hasler, May 04 2015

More terms from M. F. Hasler, May 04 2015

A263205 Start of a string of exactly 8 consecutive (but disjoint) pairs of twin primes.

Original entry on oeis.org

1107819732821, 3735283249697, 4588646146631, 6340698579419, 8412649748537, 9206359843907, 9667145661911, 10261787848841, 10877306469737, 13792968231041, 17231043159311, 18996369140627, 21471510972419, 21791129807147, 23105869316669, 23224938371519

Offset: 1

Dmitry Petukhov, Oct 12 2015

nonn

			Starting from 1107819732769 = A151799(A151799(1107819732821)), the gaps between the next primes are (40, 12, 2, 88, 2, 4, 2, 28, 2, 10, 2, 16, 2, 58, 2, 22, 2, 24, 16) with 8 occurrences of 2, so 1107819732821 is a term. - _Michel Marcus_, Oct 16 2015

Tomáš Brada, Table of n, a(n) for n = 1..10000 (first 167 terms from Dmitry Petukhov)

Cf. A001359, A035789, A035790, A035791, A035792, A035793, A035794, A035795, A087641.

cp's OEIS Frontend

A035789 Start of a string of exactly 1 consecutive (but disjoint) pair of twin primes.

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A035790 Start of a string of exactly 2 consecutive (but disjoint) pairs of twin primes.

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A035791 Start of a string of exactly 3 consecutive (but disjoint) pairs of twin primes.

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A035792 Start of a string of exactly 4 consecutive (but disjoint) pairs of twin primes.

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A035795 Start of a string of exactly 7 consecutive (but disjoint) pairs of twin primes.

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A035793 Start of a string of exactly 5 consecutive (but disjoint) pairs of twin primes.

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A035794 Start of a string of exactly 6 consecutive (but disjoint) pairs of twin primes.

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A348168 Segment the list of prime numbers into sublists L_1, L_2, ... with L_1 = {2} and L_n = {p_1, p_2, ..., p_a(n)}, where a(n) is the largest m such that for 0 < i < m, p_1 - prevprime(p_1) > p_2 - p_1 >= p_{i+1} - p_i.

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