A111950 -id:A111950 - cp's OEIS frontend

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

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

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

A338386 The smallest number from the n-membered group of single (non-twin) primes.

Original entry on oeis.org

23, 47, 79, 79, 353, 353, 353, 353, 353, 353, 673, 673, 673, 673, 673, 673, 673, 673, 8641, 8641, 8641, 8641, 13411, 13411, 13411, 14633, 14633, 24439, 24439, 24439, 24439, 24439, 24439, 24439, 24439, 24439, 24439, 24439, 24439, 24439, 62303, 62303, 62303, 62303

Offset: 1

Todor Szimeonov, Oct 23 2020

nonn

Note that "single" means both non-twin and not 2.

Todor Szimeonov, Primes - four conjectures

Cf. A007510, A111950.

c = cm = s1 = 0; p = 3; q = 5; s = {}; Do[If[c == 0, s1 = q]; r = NextPrime[q]; If[r > q + 2 && q > p + 2, c++, c = 0]; If[c > cm, cm = c; AppendTo[s, s1]]; p = q; q = r, {10^4}]; s (* Amiram Eldar, Oct 25 2020 *)

More terms from Amiram Eldar, Oct 25 2020

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

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A179067 Orders of consecutive clusters of twin primes.

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A338386 The smallest number from the n-membered group of single (non-twin) primes.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions