A261533 Primes p such that p+2 is prime with prime(p+2)-prime(p)=6.

3, 5, 59, 2789, 5231, 6947, 8087, 11717, 15269, 16229, 17207, 17909, 18059, 18131, 24917, 28751, 35279, 37307, 39227, 39239, 41201, 43787, 45821, 47741, 51869, 53087, 53609, 58439, 64577, 69857, 70919, 75707, 79631, 84869, 92381, 93479, 96179, 102197, 102929, 106187

Offset

1,1

Comments

The conjecture in A261528 implies that the current sequence has infinitely many terms.

Note that for each n > 2 the difference prime(n+2)-prime(n) is at least 6.

References

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Example

a(1) = 3 since 3 and 3+2 = 5 are twin prime, and prime(5)-prime(3) = 11-5 = 6.
a(2) = 5 since 5 and 5+2 = 7 are twin prime, and prime(7)-prime(5) = 17-11 = 6.

Mathematica

f[n_]:=Prime[n]
PQ[k_]:=PrimeQ[f[k]+2]&&f[f[k]+2]-f[f[k]]==6
n=0; Do[If[PQ[k], n=n+1; Print[n, " ", f[k]]], {k, 1, 10119}]
Select[Partition[Prime[Range[11000]], 2, 1], #[[2]]-#[[1]]==2&&Prime[#[[1]]+ 2]- Prime[#[[1]]]==6&][[All, 1]] (* Harvey P. Dale, Apr 26 2020 *)

Prog

(PARI) isok(i)=p=prime(i); isprime(p+2)&&prime(p+2)-prime(p)==6;
first(m)=my(v=vector(m)); i=1; for(j=1, m, while(!isok(i), i++); v[j]=prime(i); i++); v; \\ Anders Hellström, Aug 23 2015

Crossrefs

Cf. A000040, A001359, A006512, A023201, A046117, A236458, A259488, A259539, A261528.

Sequence in context: A174920 A158314 A249011 * A118477 A320082 A273254

Adjacent sequences: A261530 A261531 A261532 * A261534 A261535 A261536

Keyword

nonn

Author

Zhi-Wei Sun, Aug 23 2015

Status

approved