A035789 -id:A035789 - cp's OEIS frontend

A069454 Duplicate of A035789.

Original entry on oeis.org

29, 41, 59, 71, 227, 239, 269, 281, 311, 347, 461, 521, 569, 599, 617, 641, 659, 857

Offset: 1

dead

A087641 Start of the first sequence of exactly n consecutive pairs of twin primes.

Original entry on oeis.org

29, 101, 5, 9419, 909287, 325267931, 678771479, 1107819732821, 170669145704411, 3324648277099157, 789795449254776509

Offset: 1

Hugo Pfoertner, Sep 15 2003

Start of the smallest twin prime clusters of order n such that the following and preceding two primes must be neither twin primes between themselves nor with the ends of the string. - Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 22 2006

Sequences of n consecutive pairs of twin primes are called twin prime clusters of order n. Here (and in the sequences A035789, ..., A035795) it is requested that the order be exactly n, i.e., the preceding prime and the following prime must not be (upper resp. smaller) member of another twin prime pair. Note that a(3)=5 is preceded by 3 which is member of the twin prime pair (3,5) but not upper member of a preceding twin prime pair. Since it cannot happen elsewhere that P2=P3-2 if P3=P4-2 (using notations of A179067 and A035791), there is no condition imposed on P3-P2, and the condition on P2-P1 is also satisfied for P3=5. This sequence lists the starting prime of the cluster corresponding to the first occurrence of n in A179067. - M. F. Hasler, May 04 2015

			a(6)=325267931 is the starting point of the first occurrence of 6 consecutive pairs of twin primes: (325267931 325267933) (325267937 325267939) (325267949 325267951) (325267961 325267963) (325267979 325267981) (325267991 325267993).

Hugo Pfoertner, Consecutive pairs of twin primes. FORTRAN program.
Carlos Rivera, Puzzle 122. Consecutive Twin primes, The Prime Puzzles and Problems Connection.
Eric Weisstein's World of Mathematics, Twin Prime Cluster

Cf. A001359, A111950.

The sequence consists of the initial terms of A035789, A035790, A035791, A035792, A035793, A035794, A035795, A263205, A259034.

Extended by Jud McCranie
a(8)-a(10) from Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 22 2006
a(11) found by Gabor Levai in October 2011 (see Rivera), added by Dmitry Kamenetsky, Dec 15 2018

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

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

A069453 Lonely twin primes.

Original entry on oeis.org

29, 31, 41, 43, 59, 61, 71, 73, 227, 229, 239, 241, 269, 271, 281, 283, 311, 313, 347, 349, 461, 463, 521, 523, 569, 571, 599, 601, 617, 619, 641, 643, 659, 661, 857, 859, 881, 883, 1091, 1093, 1151, 1153, 1229, 1231, 1277, 1279, 1289

Offset: 1

Neil Fernandez, Mar 22 2002

nonn

			The first twin primes that are not adjacent to twin primes other than their own twins are 29,31 (23 and 37 are non-twin) and 41,43 (37 and 47 are non-twin). So the sequence begins 29,31,41,43.

Cf. A001097, A035789, A069455.

a069453[n_] := Module[{c6p={3, 5, 7, 11, 13, 17}, seq={}, i=6, last=17}, While[i<=n, If[c6p[[4]]-c6p[[3]]==2 && c6p[[2]]-c6p[[1]]>2 && c6p[[6]]-c6p[[5]]>2, AppendTo[seq, {c6p[[3]], c6p[[4]]}]]; last=NextPrime[last]; c6p=Append[Drop[c6p, 1], last]; i++]; seq]
Flatten[a069453[1000]] (* isolated prime pairs less than 1000 *)
(* Hartmut F. W. Hoft, Apr 09 2016 *)

a(2n-1) = A035789(n) and a(2n) = A069455(n), for n>=1. - Hartmut F. W. Hoft, Apr 09 2016

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

A069454 Duplicate of A035789.

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A087641 Start of the first sequence of exactly n consecutive pairs 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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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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A069453 Lonely twin primes.

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A179067 Orders of consecutive clusters of twin primes.

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