A391028 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.

2, 3, 3, -1, 3, 3, 19, 5, 23, 17, 13, 3, 7

Offset

0,1

Comments

Presumably all terms are prime numbers, if they exist.

Links

Table of n, a(n) for n=0..12.

Example

a(0) = 2 because ((2^0+1)^2-1)/2^0 = 3 (prime);
a(1) = 3 because ((2^1+1)^3-1)/2^1 = 13 (prime);
a(2) = 3 because ((2^2+1)^3-1)/2^2 = 43 (prime);
a(4) = 3 because ((2^4+1)^3-1)/2^4 = 307 (prime);
a(5) = 3 because ((2^5+1)^3-1)/2^5 = 1123 (prime).

Mathematica

a[n_]:=Module[{k=1}, If[n==3, -1, While[!PrimeQ[((2^n+1)^k-1)/(2^n)], k++]; k]]; Array[a, 13, 0] (* James C. McMahon, Dec 03 2025 *)

Prog

(Magma) [2, 3, 3, -1] cat [Min([k: k in [1..100] | ((2^n+1)^k-1) mod 2^n eq 0 and IsPrime(((2^n+1)^k-1) div 2^n)]): n in [4..12]];
(PARI) a(n) = if(n==3, return(-1)); my(k=1); while (!isprime(((2^n+1)^k - 1)/2^n), k++); k; \\ Michel Marcus, Nov 26 2025

Crossrefs

Cf. A000051, A000079, A246005.

Primes p such that ((2^m+1)^p - 1)/2^m is prime: A000043 (m = 0), A028491 (m = 1), A004061 (m = 2), no sequence in case m = 3 (see comment in A002452), A006034 (m = 4), A209120 (m = 5).

Sequence in context: A279813 A256909 A343533 * A308734 A279004 A172528

Adjacent sequences: A391025 A391026 A391027 * A391029 A391030 A391031

Keyword

sign,more

Author

Juri-Stepan Gerasimov, Nov 25 2025

Status

approved