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

%I A046135 #23 Jun 26 2025 16:31:11
%S A046135 5,11,17,29,41,59,71,101,137,179,227,239,269,281,347,419,431,641,809,
%T A046135 827,1019,1049,1091,1151,1277,1289,1427,1481,1487,1607,1697,1721,1877,
%U A046135 2027,2087,2129,2141,2339,2381,2687,2729,2789,2999,3359,3527,3581
%N A046135 Primes p such that p+2 and p+12 are primes.
%C A046135 From _Jonathan Vos Post_, May 17 2006: (Start)
%C A046135 Could be defined as "Numbers n such that k^3+k^2+n is prime for k = 0, 1, 2."
%C A046135 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)
%H A046135 Vincenzo Librandi, <a href="/A046135/b046135.txt">Table of n, a(n) for n = 1..1000</a>
%H A046135 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeTriplet.html">Prime Triplet</a>
%F A046135 {n such that n prime, n+2 prime, n+12 prime} = A001359 INTERSECT A046133. - Jonathan Vos Post, May 17 2006
%t A046135 Select[Prime[Range[600]], PrimeQ[# + 2] && PrimeQ[# + 12]&] (* _Vincenzo Librandi_, Apr 09 2013 *)
%t A046135 Select[Prime[Range[600]],AllTrue[#+{2,12},PrimeQ]&] (* _Harvey P. Dale_, Jun 26 2025 *)
%o A046135 (Magma) [p: p in PrimesUpTo(3600) | IsPrime(p+2) and IsPrime(p+12)]; // _Vincenzo Librandi_, Apr 09 2013
%Y A046135 Cf. A000040, A001359, A046133, A050266.
%K A046135 nonn,easy
%O A046135 1,1
%A A046135 _Eric W. Weisstein_
%E A046135 Edited by R. J. Mathar and _N. J. A. Sloane_, Aug 13 2008