A046135 Primes p such that p+2 and p+12 are primes.

5, 11, 17, 29, 41, 59, 71, 101, 137, 179, 227, 239, 269, 281, 347, 419, 431, 641, 809, 827, 1019, 1049, 1091, 1151, 1277, 1289, 1427, 1481, 1487, 1607, 1697, 1721, 1877, 2027, 2087, 2129, 2141, 2339, 2381, 2687, 2729, 2789, 2999, 3359, 3527, 3581

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Comments

From Jonathan Vos Post, May 17 2006: (Start)

Could be defined as "Numbers n such that k^3+k^2+n is prime for k = 0, 1, 2."

The following subset is also prime for k = 3: 5, 11, 17, 71, 101, 137, 227, 281, 347, 431, 641, 827, 1151, 1277, 1487. The following subset of those is also prime for k = 4: 17, 71, 101, 227, 827, 1151, 1487. The following subset of those is also prime for k = 5: 827, 1151, 1487. The "17" in A050266's n^3+n^2+17 is because k^3+k^2+17 is prime for k = 1,2,3,4,5,6,7,8,9,10. Between 10000 and 20000 there are 30 members of the k = 0,1,2 sequence, of which these 10 are also prime for k = 3: 10301, 10937, 11057, 11777, 12107, 13997, 15137, 15737, 16061, 19541. The following subset of those is also prime for k = 5: 15137, 15737, 16061. Somewhere in these sequences is a value that breaks the 11-term record of A050266 and indeed any known prime generating polynomial record. (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Prime Triplet

Crossrefs

Cf. A000040, A001359, A046133, A050266.

Programs

Magma

[p: p in PrimesUpTo(3600) | IsPrime(p+2) and IsPrime(p+12)]; // Vincenzo Librandi, Apr 09 2013

Mathematica

Select[Prime[Range[600]], PrimeQ[# + 2] && PrimeQ[# + 12]&] (* Vincenzo Librandi, Apr 09 2013 *)
Select[Prime[Range[600]],AllTrue[#+{2,12},PrimeQ]&] (* Harvey P. Dale, Jun 26 2025 *)

Formula

{n such that n prime, n+2 prime, n+12 prime} = A001359 INTERSECT A046133. - Jonathan Vos Post, May 17 2006

Extensions

Edited by R. J. Mathar and N. J. A. Sloane, Aug 13 2008