A389257 - OEIS

%I #35 Dec 09 2025 22:38:02

%S 3,3,3,109,3,317,7,331,3,29,53

%N a(n) is the smallest integer k such that ((2^n - 1)^k + 1)/2^n is prime, or -1 if no such k exists.

%C All terms are prime numbers, if they exist.

%e a(2) = 3 because ((2^2 - 1)^3 + 1)/2^2 = 7 (prime),

%e a(3) = 3 because ((2^3 - 1)^3 + 1)/2^3 = 43 (prime),

%e a(4) = 3 because ((2^4 - 1)^3 + 1)/2^4 = 211 (prime).

%t a[n_]:=Module[{k=1}, While[!PrimeQ[ ((2^n - 1)^k + 1)/2^n],k++];k];Array[a,11,2] (* _James C. McMahon_, Dec 09 2025 *)

%o (Magma) [Min([k: k in [1..317] | ((2^n-1)^k+1) mod 2^n eq 0 and IsPrime(((2^n-1)^k+1) div 2^n)]): n in [2..7]];

%o (PARI) isok(n,k) = my(z=((2^n - 1)^k + 1)/2^n); (denominator(z)==1) && ispseudoprime(z);

%o a(n) = my(k=1); while (!isok(n,k), k++); k; \\ _Michel Marcus_, Dec 03 2025

%Y Cf. A000225, A084742, A389827, A389883, A391028.

%Y Primes p such that ((2^m - 1)^p + 1)/2^m is prime: A007658 (m=2), A057173 (m=3), A057181 (m=4), A126856 (m=5).

%K nonn,more

%O 2,1

%A _Juri-Stepan Gerasimov_, Dec 02 2025