|
%I #32 Oct 03 2022 04:19:19
%S 8,50,2402,5764802,33232930569602,1104427674243920646305299202,
%T 1219760487635835700138573862562971820755615294131238402
%N Generalized Fermat numbers: 7^(2^n)+1, n >= 0.
%C From _Daniel Forgues_, Jun 19 2011: (Start)
%C Generalized Fermat numbers F_n(a) := F_n(a,1) = a^(2^n)+1, a >= 2, n >= 0, can't be prime if a is odd (as is the case for the current sequence) (Ribenboim (1996)).
%C All factors of generalized Fermat numbers F_n(a,b) := a^(2^n)+b^(2^n), a >= 2, n >= 0, are of the form k*2^m+1, k >= 1, m >=0 (Riesel (1994, 1998)). (This only expresses that the factors are odd, which means that it only applies to odd generalized Fermat numbers.) (End)
%H Vincenzo Librandi, <a href="/A078304/b078304.txt">Table of n, a(n) for n = 0..12</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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GeneralizedFermatNumber.html">Generalized Fermat Number</a>.
%H OEIS Wiki, <a href="/wiki/Generalized_Fermat_numbers">Generalized Fermat numbers</a>.
%F a(0) = 8, a(n)=(a(n-1)-1)^2+1, n >= 1.
%F a(n) = 6*a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get 6*(empty product, i.e., 1)+ 2 = 8 = a(0). This means that the GCD of any pair of terms is 2. - _Daniel Forgues_, Jun 20 2011
%F Sum_{n>=0} 2^n/a(n) = 1/6. - _Amiram Eldar_, Oct 03 2022
%e a(0) = 7^1+1 = 8 = 6*(1)+2 = 6*(empty product)+2.
%e a(1) = 7^2+1 = 50 = 6*(8)+2.
%e a(2) = 7^4+1 = 2402 = 6*(8*50)+2.
%e a(3) = 7^8+1 = 5764802 = 6*(8*50*2402)+2.
%e a(4) = 7^16+1 = 33232930569602 = 6*(8*50*2402*5764802)+2.
%e a(5) = 7^32+1 = 1104427674243920646305299202 = 6*(8*50*2402*5764802*33232930569602)+2.
%t Table[7^2^n + 1, {n, 0, 6}] (* _Arkadiusz Wesolowski_, Nov 02 2012 *)
%o (Magma) [7^(2^n) + 1: n in [0..8]]; // _Vincenzo Librandi_, Jun 20 2011
%Y Cf. A000215 (Fermat numbers: 2^(2^n)+1, n >= 0).
%Y Cf. A059919, A199591, A078303, A152581, A080176, A199592, A152585.
%K nonn,easy
%O 0,1
%A _Eric W. Weisstein_, Nov 21 2002
%E Edited by _Daniel Forgues_, Jun 19 2011
| 