A268212 - OEIS

%I #19 Sep 08 2022 08:46:15

%S 3,5,9,17,65,1025,65537,16777217,67108865,34359738369,

%T 4503599627370497,36028797018963969,39614081257132168796771975169,

%U 22300745198530623141535718272648361505980417

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

%C Conjecture: subsequence of prime terms (3, 5, 17, 65537, ...) is not the same as A249759.

%C Corresponding values of numbers k are in A098855 (numbers n such that 4^n + 2^n - 1 is prime).

%C Corresponding values of primes q: 5, 19, 71, 271, 4159, 1049599, 4295032831, ...

%C 4 out of 5 known Fermat primes (3, 5, 17, 65537) are terms; corresponding values of primes q: 5, 19, 271, 4295032831.

%e 17  = 2^4 + 1 is a term because 17^2 - 17 - 1 = 271 (prime).

%t 2^# + 1 &@ Select[Range[0, 300], PrimeQ[#^2 - # - 1 &@ (2^# + 1)] &] (* _Michael De Vlieger_, Jan 29 2016 *)

%o (Magma) [2^n + 1: n in [0..300] | IsPrime((2^n + 1)^2 - 2^n - 2)]

%o (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 Intersection of A002328 and A000051.

%Y Cf. A019434, A091567, A098855, A249759.

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Jan 28 2016