login
a(0)=1, a(n) = a(n-1)*a(n-1) + 2.
2

%I #22 Mar 27 2023 09:01:37

%S 1,3,11,123,15131,228947163,52416803445748571,

%T 2747521283470239265968814548542043,

%U 7548873203121950871924356140057489033996373873303512592376938613851

%N a(0)=1, a(n) = a(n-1)*a(n-1) + 2.

%C The Mandelbrot-process is z:=z*z+c, where z and c is complex. In our case c=2 and the initial z is 1. The process is very quickly increasing.

%C Prime for a(1)=3, a(2)=11, a(4)=15131; semiprime for a(3) = 123 = 3 * 41, a(5) = 228947163 = 3 * 76315721. a(6), added by Jonathan Vos Post, has 4 prime factors. a(7) = 41 * 811^2 * 106693969 * 317171188688357726699 * 8272236925540996054440172449761. When is the next prime in the sequence? - _Jonathan Vos Post_, Feb 28 2005

%C Composite for a(8), a(9), ..., a(19). a(20) is roughly 2^909982 and its primality is unknown. - _Russ Cox_, Apr 02 2006

%F a(n) ~ c^(2^n), where c = 1.8249111600523655937123650418390169034... - _Vaclav Kotesovec_, Sep 20 2013

%e a(2)=11, a(3)=11*11+2=123.

%p a[0]:=1: for n from 1 to 10 do a[n]:=a[n-1]^2+2 od: seq(a[n],n=0..9); # _Emeric Deutsch_

%t a[0] := 1; a[n_] := a[n - 1]^2 + 2; Table[a[n], {n, 0, 10}] (* _Stefan Steinerberger_, Apr 08 2006 *)

%t NestList[#^2+2&,1,10] (* _Harvey P. Dale_, Mar 27 2023 *)

%o (PARI) a(n)=if(n<1, n==0, 2+a(n-1)^2) /* _Michael Somos_, Mar 25 2006 */

%Y Bisection of A065653.

%K easy,nonn

%O 0,2

%A _Miklos Kristof_, Feb 28 2005

%E a(7) from _Jonathan Vos Post_, Feb 28 2005

%E a(8) from _Emeric Deutsch_, Jun 13 2005