cp's OEIS Frontend

A268212 Numbers n of the form 2^k + 1 such that n^2 - n - 1 is a prime q (for k >= 0).

%I A268212 #19 Sep 08 2022 08:46:15
%S A268212 3,5,9,17,65,1025,65537,16777217,67108865,34359738369,
%T A268212 4503599627370497,36028797018963969,39614081257132168796771975169,
%U A268212 22300745198530623141535718272648361505980417
%N A268212 Numbers n of the form 2^k + 1 such that n^2 - n - 1 is a prime q (for k >= 0).
%C A268212 Conjecture: subsequence of prime terms (3, 5, 17, 65537, ...) is not the same as A249759.
%C A268212 Corresponding values of numbers k are in A098855 (numbers n such that 4^n + 2^n - 1 is prime).
%C A268212 Corresponding values of primes q: 5, 19, 71, 271, 4159, 1049599, 4295032831, ...
%C A268212 4 out of 5 known Fermat primes (3, 5, 17, 65537) are terms; corresponding values of primes q: 5, 19, 271, 4295032831.
%e A268212 17  = 2^4 + 1 is a term because 17^2 - 17 - 1 = 271 (prime).
%t A268212 2^# + 1 &@ Select[Range[0, 300], PrimeQ[#^2 - # - 1 &@ (2^# + 1)] &] (* _Michael De Vlieger_, Jan 29 2016 *)
%o A268212 (Magma) [2^n + 1: n in [0..300] | IsPrime((2^n + 1)^2 - 2^n - 2)]
%o A268212 (PARI) lista(nn) = {for (k=0, nn, n = 2^k+1; if (isprime(n^2-n-1), print1(n, ", ")););} \\ _Michel Marcus_, Mar 06 2016
%Y A268212 Intersection of A002328 and A000051.
%Y A268212 Cf. A019434, A091567, A098855, A249759.
%K A268212 nonn
%O A268212 1,1
%A A268212 _Jaroslav Krizek_, Jan 28 2016