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%I #51 Feb 26 2023 01:32:45
%S 0,1,2,3,5,8,9,12,15,17,18,21,23,27,32,51,65,87,180,242,467,491,501,
%T 507,555,591,680,800,1070,1650,2813,3281,4217,5153,6287,6365,10088,
%U 10367,37035,45873,69312,102435,106380,108888,110615,281621,369581,376050,442052,621443,661478
%N Numbers k such that (2^k + 1)^2 - 2 = 4^k + 2^(k+1) - 1 is prime.
%H S. Harvey, <a href="http://harvey563.tripod.com/Carol_Kynea.txt">Carol and Kynea Primes</a>
%H M. Rodenkirch, <a href="http://www.mersenneforum.org/rogue/ckps.html">Carol and Kynea Prime Search</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime</a>
%F A093069(n) = (2^a(n) + 1)^2 - 2.
%t Flatten[Position[Table[(2^n + 1)^2 - 2, {n, 0, 10^3}], _?PrimeQ] - 1] (* _Eric W. Weisstein_, Feb 10 2016 *)
%t Select[Range[0, 5000], PrimeQ[(2^# + 1)^2 - 2] & ] (* _Vincenzo Librandi_, Feb 19 2016 *)
%o (Magma) [n: n in [0..500] | IsPrime((2^n+1)^2-2)]; // _Vincenzo Librandi_, Feb 19 2016
%o (PARI) is(n)=ispseudoprime((2^n+1)^2-2) \\ _Charles R Greathouse IV_, Feb 19 2016
%Y Cf. A091514 (primes of the form (2^n + 1)^2 - 2).
%Y Cf. A093069 (numbers of the form (2^n + 1)^2 - 2).
%K nonn,hard
%O 1,3
%A _Eric W. Weisstein_, Jan 17 2004
%E a(41) from _Eric W. Weisstein_, Feb 27 2004
%E a(42) to a(44) from _Eric W. Weisstein_, Jun 05 2004
%E Edited by _Ray Chandler_, Nov 15 2004
%E a(46) from Cletus Emmanuel (cemmanu(AT)yahoo.com), Oct 07 2005
%E a(47)-a(48) from _Eric W. Weisstein_, Feb 10 2016 (computed by Mark Rodenkirch)
%E a(49)-a(50) from _Eric W. Weisstein_, Jun 08 2016 (computed by Mark Rodenkirch)
%E a(51) from _Eric W. Weisstein_, Jun 19 2016 (computed by Mark Rodenkirch)
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