OFFSET
1,2
COMMENTS
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
LINKS
EXAMPLE
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)
MAPLE
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
MATHEMATICA
Length/@Split[Select[Range[2, 100], Prime[#+1]-Prime[#]==2&], #2==#1+2&] (* Gus Wiseman, Dec 05 2024 *)
PROG
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Franz Vrabec, Jun 27 2010
EXTENSIONS
More terms from M. F. Hasler, May 04 2015
STATUS
approved