A128025 - OEIS

%I #20 Jul 29 2023 21:49:10

%S 2,3,7,19,31,67,89,9227,43891,854149

%N Numbers k such that (8^k - 3^k)/5 is prime.

%C All terms are primes.

%C Verified the first 8 terms in sequence. Also, the next number in the sequence, if one exists is > 43691. - _Robert Price_, Mar 16 2010

%C a(10) > 10^5. - _Robert Price_, Jul 27 2011

%C a(11) > 10^6. - _Jon Grantham_, Jul 29 2023

%H Jon Grantham and Andrew Granville, <a href="https://arxiv.org/abs/2307.07894">Fibonacci primes, primes of the form 2^n-k and beyond</a>, arXiv:2307.07894 [math.NT], 2023.

%t k=5; Select[ Prime[ Range[1,200] ], PrimeQ[ ((k+3)^# - 3^#)/k ]& ]

%o (PARI) is(n)=isprime((8^n-3^n)/5) \\ _Charles R Greathouse IV_, Feb 17 2017

%Y Cf. A028491 = numbers n such that (3^n - 1)/2 is prime. Cf. A057468 = numbers n such that 3^n - 2^n is prime. Cf. A059801 = numbers n such that 4^n - 3^n is prime. Cf. A121877 = numbers n such that (5^n - 3^n)/2 is a prime. Cf. A128024, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

%K hard,more,nonn

%O 1,1

%A _Alexander Adamchuk_, Feb 11 2007

%E 9227 from _Farideh Firoozbakht_, Apr 08 2007

%E a(9) from _Robert Price_, Jul 27 2011

%E a(10) from _Jon Grantham_, Jul 29 2023