A389257 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.

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

Offset

2,1

Comments

All terms are prime numbers, if they exist.

Links

Table of n, a(n) for n=2..12.

Example

a(2) = 3 because ((2^2 - 1)^3 + 1)/2^2 = 7 (prime),
a(3) = 3 because ((2^3 - 1)^3 + 1)/2^3 = 43 (prime),
a(4) = 3 because ((2^4 - 1)^3 + 1)/2^4 = 211 (prime).

Mathematica

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 *)

Prog

(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]];
(PARI) isok(n, k) = my(z=((2^n - 1)^k + 1)/2^n); (denominator(z)==1) && ispseudoprime(z);
a(n) = my(k=1); while (!isok(n, k), k++); k; \\ Michel Marcus, Dec 03 2025

Crossrefs

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

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).

Sequence in context: A217671 A118539 A015665 * A343120 A230667 A131462

Adjacent sequences: A389254 A389255 A389256 * A389258 A389259 A389260

Keyword

nonn,more

Author

Juri-Stepan Gerasimov, Dec 02 2025

Status

approved