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Smallest prime of form 2^n*k + 1.
12

%I #43 Jun 21 2023 12:06:55

%S 2,3,5,17,17,97,193,257,257,7681,12289,12289,12289,40961,65537,65537,

%T 65537,786433,786433,5767169,7340033,23068673,104857601,167772161,

%U 167772161,167772161,469762049,2013265921,3221225473,3221225473,3221225473,75161927681

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

%C 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

%H Alois P. Heinz, <a href="/A035089/b035089.txt">Table of n, a(n) for n = 0..1000</a>

%H Gareth A. Jones and Alexander K. Zvonkin, <a href="https://www.labri.fr/perso/zvonkin/Research/ProjPrimesShort.pdf">Groups of prime degree and the Bateman-Horn conjecture</a>, 2021.

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

%o (PARI) a(n)=for(k=1,9e99,if(ispseudoprime(k<<n+1),return(k<<n+1))) \\ _Charles R Greathouse IV_, Jul 06 2011

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

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

%K nonn

%O 0,1

%A _Labos Elemer_

%E a(0) from _Joerg Arndt_, Jul 06 2011