A105413 Numbers p(n) such that both p(n)+2 and p(n+6)-2 are prime numbers, where p(n) is the n-th prime.

3, 11, 107, 239, 311, 569, 1019, 1031, 1229, 1427, 1997, 2081, 2087, 2111, 2687, 3251, 4049, 4127, 4157, 4229, 4241, 4481, 5231, 5639, 6089, 7307, 7559, 8969, 9629, 10007, 10457, 13691, 13829, 13901, 14249, 14549, 14561, 16187, 16649, 17207

Offset

1,1

Comments

Conjecture: There are an infinite number of primes p(n) such that p(n)+2 and p(n+k)-2 are both prime for all k > 1.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Example

prime(5)=11, and both prime(5)+2=13 and prime(5+6)-2=29 are prime, so 11 is in the sequence.

Mathematica

For[n = 1, n < 500, n++, If[PrimeQ[Prime[n] + 2], If[PrimeQ[Prime[n + 6] - 2], Print[Prime[n]]]]] (* Stefan Steinerberger, Feb 07 2006 *)
Transpose[Select[Partition[Prime[Range[2000]], 7, 1], #[[2]]-#[[1]] == #[[7]]- #[[6]] == 2&]][[1]] (* Harvey P. Dale, Oct 08 2014 *)

Prog

(PARI) pnpk(n, m=6, k=2) = { local(x, v1, v2); for(x=1, n, v1 = prime(x)+ k; v2 = prime(x+m)-k; if(isprime(v1)&isprime(v2), print1(prime(x), ", ") ) ) ; } \\ corrected by Michel Marcus, Sep 14 2015
(Magma) [NthPrime(n): n in [1..2000] | IsPrime(NthPrime(n)+2) and IsPrime(NthPrime(n+6)-2)]; // Vincenzo Librandi, Sep 14 2015

Crossrefs

Cf. A089635. - Harvey P. Dale, Oct 08 2014

Sequence in context: A238446 A337734 A302927 * A241587 A183381 A136985

Adjacent sequences: A105410 A105411 A105412 * A105414 A105415 A105416

Keyword

nonn

Author

Cino Hilliard, May 02 2005

Status

approved