A069455 -id:A069455 - cp's OEIS frontend

A035789 Start of a string of exactly 1 consecutive (but disjoint) pair of twin primes.

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

Randall L Rathbun, Nov 30 1998

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.

			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)

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000
Hugo Pfoertner, FORTRAN program: Consecutive pairs of twin primes.
Randall Rathbun, A study of n-twin_prime clusters among prime numbers, Posting to Number Theory List, Nov 19 1998.

Cf. A035790, A035791, A035792, A035793, A035794, A035795, A087641.

Cf. A069453, A069455.

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 *)

Edited by Hugo Pfoertner, Oct 15 2003

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

A262935 Increasing distances of lonely twin primes pairs to nearest prime.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 18, 28, 30, 34, 42, 46, 48, 58, 88, 90, 94, 124, 130, 136, 154, 162, 168, 172, 178, 202, 216, 258, 264, 294, 342, 352, 354, 364, 366, 370, 378, 396, 408

Offset: 1

Dmitry Petukhov, Oct 04 2015

nonn

			(3,5) is a twin primes pair, min(7-5, 3-2)=1, therefore a(1)=1.
(5,7) is a twin primes pair, min(11-7, 5-3)=2>1, therefore a(2)=2.
(11,13) is a twin primes pair, min(17-13, 11-7)=4>2, therefore a(3)=4.

Cf. A035789, A069453, A069455, A262936.

{m=0; q=5; s=3; t=2; forprime(p=6, 10^9, if((q-s==2) && (min(p-q, s-t)>m), m=min(p-q, s-t); print1(m, ", ") ); t=s; s=q; q=p;)}

a(n) = d if ( (p(i+1) = p(i)+2) AND (d = min(p(i+2)-p(i+1), p(i)-p(i-1)) > a(n-1)) ), where a(0) = 0, p(k) = prime(k) = A000040(k).

cp's OEIS Frontend

A035789 Start of a string of exactly 1 consecutive (but disjoint) pair of twin primes.

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A069453 Lonely twin primes.

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A262935 Increasing distances of lonely twin primes pairs to nearest prime.

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