A276125 - OEIS

%I #30 May 12 2023 12:00:00

%S 1,2,3,2,6,6,12,2,24,6,30,6,30,22,6,2,30,24,90,6,33,42,59,6,40,30,72,

%T 22,77,6,73,2,66,30,66,24,72,90,60,6,99,66,122,42,48,118,144,6,432,40,

%U 60,30,132,72,66,22,129,154,870,6,210,146,264,2,60,66,224,30

%N a(n) = maximum eventual period of z := z^2 + c (mod n), for Gaussian integers z, c.

%C Note that this is the maximum over all possible initial z.

%C From _Robert Israel_, Aug 29 2016: (Start)

%C If n is divisible by 4, then a(n) = a(n/2).

%C In particular, a(n) = 2 if n > 1 is a power of 2.

%C Are there any other n with a(n) = 2? (End)

%e For n = 3, c = 2+i:

%e z_0 = 0.

%e z_1 = (z_0)^2 + 2+i = 2+ i (mod 3).

%e z_2 = (z_1)^2 + 2+i = 2+2i (mod 3).

%e z_3 = (z_2)^2 + 2+i = 2    (mod 3).

%e z_4 = (z_3)^2 + 2+i =    i (mod 3).

%e z_5 = (z_4)^2 + 2+i = 1+ i (mod 3).

%e z_6 = (z_5)^2 + 2+i = 2    (mod 3) = z_3.

%e So the eventual period is 3, which is the maximum possible over all c and z_0 when n = 3.

%p f:= proc(N)

%p   local pmax,R,S,T,z,a,b,c,x,y,found,zpd,count;

%p   pmax:= 0;

%p   for a from 0 to N-1 do

%p     for b from 0 to N-1 do

%p       c:= a+b*I;

%p       S:= {}:

%p       for x from 0 to N-1 do

%p         for y from 0 to N-1 do

%p            z:= x+I*y;

%p            if not member(z,S) then

%p              T:= {z};

%p              R[z]:= 0;

%p              found:= false;

%p              for count from 1 do

%p                z:= expand(z^2 + c) mod N;

%p                if member(z,S) then break fi;

%p                if member(z,T) then found:= true; zpd:= count - R[z]; break fi;

%p                R[z]:= count;

%p                T:= T union {z};

%p              od;

%p              S:= S union T;

%p              if found and zpd > pmax then

%p                 pmax:= zpd fi;

%p            fi;

%p       od od;

%p   od od;

%p   pmax

%p end proc:

%p map(f, [$1..30]); # _Robert Israel_, Aug 29 2016

%t f[n_] := Module[{pmax = 0, R, S, T, z, a, b, c, x, y, found, zpd, count},

%t  For[a = 0, a <= n - 1, a++,

%t    For[b = 0, b <= n - 1, b++, c = a + b*I; S = {};

%t      For[x = 0, x <= n - 1, x++,

%t        For[y = 0, y <= n - 1, y++, z = x + y*I;

%t          If[!MemberQ[S, z], T = {z}; R[z] = 0; found = False;

%t          For[count = 1, True, count++,

%t            z = Mod[Expand[z^2 + c], n];

%t            If[MemberQ[S, z], Break[] ];

%t            If [MemberQ[T, z], found = True; zpd = count -

%t            R[z]; Break[]]; R[z] = count;

%t            T = Union[T, {z}]]; S = Union[S, T];

%t            If [found && zpd > pmax, pmax = zpd]]]]]];

%t   pmax];

%t Table[f[n], {n, 1, 20}] (* _Jean-François Alcover_, May 12 2023, after _Robert Israel_ *)

%K nonn

%O 1,2

%A _Luke Palmer_, Aug 21 2016

%E More terms from _Robert Israel_, Aug 29 2016