A092506 Prime numbers of the form 2^n + 1.

2, 3, 5, 17, 257, 65537

Offset

1,1

Comments

2 together with the Fermat primes A019434.

Obviously if 2^n + 1 is a prime then either n = 0 or n is a power of 2. - N. J. A. Sloane, Apr 07 2004

Numbers m > 1 such that 2^(m-2) divides (m-1)! and m divides (m-1)! + 1. - Thomas Ordowski, Nov 25 2014

From Jaroslav Krizek, Mar 06 2016: (Start)

Also primes p such that sigma(p-1) = 2p - 3.

Also primes of the form 2^n + 3*(-1)^n - 2 for n >= 0 because for odd n, 2^n - 5 is divisible by 3.

Also primes of the form 2^n + 6*(-1)^n - 5 for n >= 0 because for odd n, 2^n - 11 is divisible by 3.

Also primes of the form 2^n + 15*(-1)^n - 14 for n >= 0 because for odd n, 2^n - 29 is divisible by 3. (End)

Exactly the set of primes p such that any number congruent to a primitive root (mod p) must have at least one prime divisor that is also congruent to a primitive root (mod p). See the links for a proof. - Rafay A. Ashary, Oct 13 2016

Conjecture: these are the only solutions to the equation A000010(x)+A000010(x-1)=floor((3x-2)/2). - Benoit Cloitre, Mar 02 2018

For n > 1, if 2^n + 1 divides 3^(2^(n-1)) + 1, then 2^n + 1 is a prime. - Jinyuan Wang, Oct 13 2018

The prime numbers occurring in A003401. Also, the prime numbers dividing at least one term of A003401. - Jeppe Stig Nielsen, Jul 24 2019

Links

Table of n, a(n) for n=1..6.

Rafay A. Ashary, A Property of A092506

Barry Brent, On the constant terms of certain meromorphic modular forms for Hecke groups, arXiv:2212.12515 [math.NT], 2022.

Eric Weisstein's World of Mathematics, Fermat Prime

Eric Weisstein's World of Mathematics, Fermat Number

Mathematica

Select[2^Range[0, 100]+1, PrimeQ] (* Harvey P. Dale, Aug 02 2015 *)

Prog

(PARI) print1(2); for(n=0, 9, if(ispseudoprime(t=2^2^n+1), print1(", "t))) \\ Charles R Greathouse IV, Aug 29 2016
(Magma) [2^n + 1 : n in [0..25] | IsPrime(2^n+1)]; // Vincenzo Librandi, Oct 14 2018
(GAP) Filtered(List([1..20], n->2^n+1), IsPrime); # Muniru A Asiru, Oct 25 2018

Crossrefs

A019434 is the main entry for these numbers.

Sequence in context: A099936 A237251 A275584 * A275159 A127063 A127837

Adjacent sequences: A092503 A092504 A092505 * A092507 A092508 A092509

Keyword

nonn,hard

Author

Jorge Coveiro, Apr 05 2004

Status

approved