|
%I #81 Dec 28 2022 17:57:26
%S 2,3,5,17,257,65537
%N Prime numbers of the form 2^n + 1.
%C 2 together with the Fermat primes A019434.
%C 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
%C Numbers m > 1 such that 2^(m-2) divides (m-1)! and m divides (m-1)! + 1. - _Thomas Ordowski_, Nov 25 2014
%C From _Jaroslav Krizek_, Mar 06 2016: (Start)
%C Also primes p such that sigma(p-1) = 2p - 3.
%C 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.
%C 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.
%C 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)
%C 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
%C Conjecture: these are the only solutions to the equation A000010(x)+A000010(x-1)=floor((3x-2)/2). - _Benoit Cloitre_, Mar 02 2018
%C 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
%C The prime numbers occurring in A003401. Also, the prime numbers dividing at least one term of A003401. - _Jeppe Stig Nielsen_, Jul 24 2019
%H Rafay A. Ashary, <a href="/A092506/a092506.pdf">A Property of A092506</a>
%H Barry Brent, <a href="https://arxiv.org/abs/2212.12515">On the constant terms of certain meromorphic modular forms for Hecke groups</a>, arXiv:2212.12515 [math.NT], 2022.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FermatPrime.html">Fermat Prime</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FermatNumber.html">Fermat Number</a>
%t Select[2^Range[0,100]+1,PrimeQ] (* _Harvey P. Dale_, Aug 02 2015 *)
%o (PARI) print1(2); for(n=0,9, if(ispseudoprime(t=2^2^n+1), print1(", "t))) \\ _Charles R Greathouse IV_, Aug 29 2016
%o (Magma) [2^n + 1 : n in [0..25] | IsPrime(2^n+1)]; // _Vincenzo Librandi_, Oct 14 2018
%o (GAP) Filtered(List([1..20],n->2^n+1),IsPrime); # _Muniru A Asiru_, Oct 25 2018
%Y A019434 is the main entry for these numbers.
%K nonn,hard
%O 1,1
%A _Jorge Coveiro_, Apr 05 2004
| 