A179067 - OEIS

%I #16 Dec 05 2024 22:42:28

%S 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,

%T 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,

%U 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

%N Orders of consecutive clusters of twin primes.

%C 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.

%C 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

%H Robert Israel, <a href="/A179067/b179067.txt">Table of n, a(n) for n = 1..10000</a>

%H Gus Wiseman, <a href="/A373403/a373403.txt">Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers</a>.

%e 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.

%e From _Gus Wiseman_, Dec 05 2024: (Start)

%e The leading indices of twin primes are:

%e   2, 3, 5, 7, 10, 13, 17, 20, 26, 28, 33, 35, 41, 43, 45, 49, 52, ...

%e with maximal runs of terms differing by 2:

%e   {2}, {3,5,7}, {10}, {13}, {17}, {20}, {26,28}, {33,35}, {41,43,45}, {49}, {52}, ...

%e with lengths a(n).

%e (End)

%p R:= 1: count:= 1: m:= 0:

%p q:= 5: state:= 1:

%p while count < 100 do

%p  p:= nextprime(q);

%p  if state = 1 then

%p     if p-q = 2 then state:= 2; m:= m+1;

%p     else

%p       if m > 0 then R:= R,m; count:= count+1; fi;

%p       m:= 0

%p     fi

%p  else state:= 1;

%p  fi;

%p  q:= p

%p od:

%p R; # _Robert Israel_, Feb 07 2023

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

%o (PARI) 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

%Y Cf. A077800.

%Y Cf. A001359, A111950, A087641.

%Y Cf. A035789, A035790, A035791, A035792, A035793, A035794, A035795.

%Y A000040 lists the primes, differences A001223 (run-lengths A333254, A373821).

%Y A006512 gives the greater of twin primes.

%Y A029707 gives the leading index of twin primes, complement A049579.

%Y A038664 finds the first prime gap of length 2n.

%Y A046933 counts composite numbers between primes.

%Y A000720, A006560, A006562, A014574, A037201, A107770, A122535, A155752, A175632, A251092.

%K easy,nonn

%O 1,2

%A _Franz Vrabec_, Jun 27 2010

%E More terms from _M. F. Hasler_, May 04 2015