A259539 Numbers m with m-1, m+1 and prime(m)+2 all prime.

60, 828, 858, 1032, 1050, 1230, 1320, 1878, 2028, 2340, 3252, 3390, 3462, 4548, 5502, 6870, 6948, 7590, 7878, 8010, 9438, 9720, 9858, 10038, 10068, 10302, 11490, 11718, 13932, 14388, 15138, 15270, 15288, 16068, 16188, 16230, 17208, 17292, 17838, 17910

Offset: 1

Author

Zhi-Wei Sun, Jun 30 2015

Keywords

• nonn

Comments

Conjecture: The sequence contains infinitely many terms.

This is stronger than the Twin Prime Conjecture, and weaker than the conjecture in A259540.

Subsequence of A014574.

Examples

			a(1) = 60 since 60-1 = 59, 60+1 = 61 and prime(60)+2 = 283 are all prime.
		

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..10000
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A001359, A006512, A014574, A236458, A259540.

Programs

Mathematica

n=0;Do[If[PrimeQ[k-1]&&PrimeQ[k+1]&&PrimeQ[Prime[k]+2],n=n+1;Print[n," ",k]],{k,1,18000}]

PARI

lista(nn) = {forprime(p=2, nn, if (isprime(p+2) && isprime(prime(p+1)+2), print1(p+1, ", ")));} \\ Michel Marcus, Jun 30 2015