cp's OEIS Frontend

A332061 Number of iterations of z -> z^2 + 1/4 + 1/n until z >= 2, starting with z = 0.

%I A332061 #13 Feb 29 2020 10:02:40
%S A332061 2,3,4,5,6,6,7,7,8,8,9,9,10,10,10,11,11,12,12,12,13,13,13,14,14,14,15,
%T A332061 15,15,15,16,16,16,17,17,17,17,18,18,18,18,19,19,19,19,20,20,20,20,20,
%U A332061 21,21,21,21,21,22,22,22,22,23,23,23,23,23,24,24,24,24,24,24,25,25,25
%N A332061 Number of iterations of z -> z^2 + 1/4 + 1/n until z >= 2, starting with z = 0.
%C A332061 The iterated map is of the form of the maps f_c: z -> z^2 + c used to define the Mandelbrot set as those complex c for which the trajectory of 0 under f_c will never leave the ball of radius 2.
%C A332061 The largest real number in the Mandelbrot set is c = 1/4, with the trajectory of 0 going to 1/2 from the left.
%C A332061 The number of iterations N(epsilon) to reach z >= 2 for c = 1/4 + epsilon is such that N(epsilon) ~ Pi/sqrt(epsilon), see the Numberphile video.
%H A332061 Brady Haran and Holly Krieger, <a href="https://youtu.be/d0vY0CKYhPY">Pi and the Mandelbrot Set</a>, Numberphile channel on YouTube, Oct. 1, 2015.
%F A332061 a(n) ~ Pi*sqrt(n), asymptotically.
%t A332061 Table[-1 + Length@ NestWhileList[#^2 + 1/4 + 1/n &, 0, # < 2 &], {n, 73}] (* _Michael De Vlieger_, Feb 25 2020 *)
%o A332061 (PARI) apply( {A332061(n,z,k)=n=.25+1/n;until(2<z=z^2+n,k++);k}, [1..99])
%o A332061 (Python)
%o A332061 def A332061(n):
%o A332061     c=1/4+1/n; z=c; n=1
%o A332061     while z<2: z=z**2+c; n+=1
%o A332061     return n
%K A332061 nonn
%O A332061 1,1
%A A332061 _M. F. Hasler_, Feb 22 2020