A035089 Smallest prime of form 2^n*k + 1.

2, 3, 5, 17, 17, 97, 193, 257, 257, 7681, 12289, 12289, 12289, 40961, 65537, 65537, 65537, 786433, 786433, 5767169, 7340033, 23068673, 104857601, 167772161, 167772161, 167772161, 469762049, 2013265921, 3221225473, 3221225473, 3221225473, 75161927681

Offset: 0

Author

Labos Elemer

Keywords

• nonn

Comments

a(n) is the smallest prime p such that the multiplicative group modulo p has a subgroup of order 2^n. - Joerg Arndt, Oct 18 2020

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Gareth A. Jones and Alexander K. Zvonkin, Groups of prime degree and the Bateman-Horn conjecture, 2021.

Crossrefs

Analogous case is A034694. Fermat primes (A019434) are a subset. See also Fermat numbers A000215.

Cf. A007522, A057775, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587.

Programs

Mathematica

a = {}; Do[k = 0; While[ !PrimeQ[k 2^n + 1], k++ ]; AppendTo[a, k 2^n + 1], {n, 1, 50}]; a (* Artur Jasinski *)

PARI

a(n)=for(k=1,9e99,if(ispseudoprime(k<Charles R Greathouse IV, Jul 06 2011

Extensions

a(0) from Joerg Arndt, Jul 06 2011