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A035789
Start of a string of exactly 1 consecutive (but disjoint) pair of twin primes.
17
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
1,1
COMMENTS
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.
LINKS
Sebastian Petzelberger, Table of n, a(n) for n = 1..10000
Randall Rathbun, A study of n-twin_prime clusters among prime numbers, Posting to Number Theory List, Nov 19 1998.
EXAMPLE
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)
MATHEMATICA
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 *)
KEYWORD
nonn,easy
AUTHOR
Randall L Rathbun, Nov 30 1998
EXTENSIONS
Edited by Hugo Pfoertner, Oct 15 2003
STATUS
approved