A152587 - OEIS

%I #30 Jun 10 2023 17:56:20

%S 15,197,38417,1475789057,2177953337809371137,

%T 4743480741674980702700443299789930497,

%U 22500609546641425009067997918450033531906583365663182830821882796510806017

%N Generalized Fermat numbers: a(n) = 14^(2^n) + 1.

%C There appears to be no divisibility rule for this sequence.

%H Charles R Greathouse IV, <a href="/A152587/b152587.txt">Table of n, a(n) for n = 0..9</a>

%H Anders Björn and Hans Riesel, <a href="http://www.jstor.org/stable/2584996">Factors of Generalized Fermat Numbers</a>, Mathematics of Computation, Vol. 67, No. 221, Jan., 1998, pp. 441-446.

%H OEIS Wiki, <a href="/wiki/Generalized_Fermat_numbers">Generalized Fermat numbers</a>

%F a(0) = 15, a(n) = (a(n-1)-1)^2 + 1, n >= 1.

%F a(n) = 13*a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get 13*(empty product, i.e., 1)+ 2 = 15 = a(0). This implies that the terms, all odd, are pairwise coprime. - _Daniel Forgues_, Jun 20 2011

%e a(0) = 14^1+1 = 15 = 13*(1)+2 = 13(empty product)+2.

%e a(1) = 14^2+1 = 197 = 13*(15)+2.

%e a(2) = 14^4+1 = 38417 = 13*(15*197)+2.

%e a(3) = 14^8+1 = 1475789057 = 13*(15*197*38417)+2.

%e a(4) = 14^16+1 = 2177953337809371137 = 13*(15*197*38417*1475789057)+2.

%e a(5) = 14^32+1 = 4743480741674980702700443299789930497 = 13*(15*197*38417*1475789057*2177953337809371137)+2.

%t Table[14^2^n+1,{n,0,7}] (* _Harvey P. Dale_, Jun 10 2023 *)

%o (PARI) g(a,n) = if(a%2,b=2,b=1);for(x=0,n,y=a^(2^x)+b;print1(y","))

%o (Magma) [14^(2^n) + 1: n in [0..8]]; // _Vincenzo Librandi_, Jun 20 2011

%o (PARI) a(n)=14^(2^n)+1 \\ _Charles R Greathouse IV_, Jul 29 2011

%Y Cf. A000215.

%K nonn,easy

%O 0,1

%A _Cino Hilliard_, Dec 08 2008