A105413 Primes p = prime(k) such that both p+2 and prime(k+6)-2 are prime numbers.

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 infinitely many primes p(k) such that p(k)+2 and p(k+m)-2 are both primes for all m > 1.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Example

prime(5) = 11, and both prime(5)+2 = 13 and prime(5+6)-2 = 29 are primes, 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
(PARI) lista(pmax) = {my(k = 1, p = primes(7)); forprime(p1 = p[#p], pmax, k++; p[#p] = p1; if(p[2]- p[1] == 2 && p[7] - p[6] == 2, print1(p[1], ", ")); for(i = 1, #p-1, p[i] = p[i+1])); } \\ Amiram Eldar, Oct 04 2024
(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, A105409, A105410, A105411, A105412, A105414.

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