A102847 a(0)=1, a(n) = a(n-1)*a(n-1) + 2.

1, 3, 11, 123, 15131, 228947163, 52416803445748571, 2747521283470239265968814548542043, 7548873203121950871924356140057489033996373873303512592376938613851

Offset: 0

Author

Miklos Kristof, Feb 28 2005

Keywords

• easy
• nonn

Comments

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.

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

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

Examples

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

Crossrefs

Bisection of A065653.

Programs

Maple

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

Mathematica

a[0] := 1; a[n_] := a[n - 1]^2 + 2; Table[a[n], {n, 0, 10}] (* Stefan Steinerberger, Apr 08 2006 *)
NestList[#^2+2&,1,10] (* Harvey P. Dale, Mar 27 2023 *)

PARI

a(n)=if(n<1, n==0, 2+a(n-1)^2) /* Michael Somos, Mar 25 2006 */

Formula

a(n) ~ c^(2^n), where c = 1.8249111600523655937123650418390169034... - Vaclav Kotesovec, Sep 20 2013

Extensions

a(7) from Jonathan Vos Post, Feb 28 2005

a(8) from Emeric Deutsch, Jun 13 2005