login
Smallest k such that 2^(2^n) + k is a safe prime.
2

%I #22 Jan 29 2023 17:52:24

%S 1,7,7,7,91,3103,12451,230191,286867,1657867,10029811,29761351,

%T 22410151,98402791,167137543

%N Smallest k such that 2^(2^n) + k is a safe prime.

%C a(n) == 3 (mod 4) for n > 1. - _Chai Wah Wu_, Jan 27 2023

%F a(n) = A350696(2^n).

%e a(3) = 7 because 2^(2^3) + 7 = 263 is the smallest safe prime greater than 256.

%o (PARI) a(n) = {my(k=1); pow2 = 2^(2^n); while (!(isprime(pow2 + k) && isprime((pow2 + k - 1)/2)), k+=2); k;} \\

%o (Python)

%o from sympy import isprime, nextprime

%o def A360080(n):

%o if n <= 1: return 1

%o m = 1<<(1<<n)-1

%o i = m

%o while i:=nextprime(i):

%o if isprime(k:=(i<<1)+1):

%o return k-(m<<1) # _Chai Wah Wu_, Jan 27 2023

%Y Cf. A360081, A181356, A005385, A058220, A013597, A013603, A335313, A350696.

%K nonn,more,hard

%O 1,2

%A _Mark Andreas_, Jan 25 2023