A276125 a(n) = maximum eventual period of z := z^2 + c (mod n), for Gaussian integers z, c.

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, 22, 77, 6, 73, 2, 66, 30, 66, 24, 72, 90, 60, 6, 99, 66, 122, 42, 48, 118, 144, 6, 432, 40, 60, 30, 132, 72, 66, 22, 129, 154, 870, 6, 210, 146, 264, 2, 60, 66, 224, 30

Offset

1,2

Comments

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

From Robert Israel, Aug 29 2016: (Start)

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

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

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

Links

Table of n, a(n) for n=1..68.

Example

For n = 3, c = 2+i:
z_0 = 0.
z_1 = (z_0)^2 + 2+i = 2+ i (mod 3).
z_2 = (z_1)^2 + 2+i = 2+2i (mod 3).
z_3 = (z_2)^2 + 2+i = 2    (mod 3).
z_4 = (z_3)^2 + 2+i =    i (mod 3).
z_5 = (z_4)^2 + 2+i = 1+ i (mod 3).
z_6 = (z_5)^2 + 2+i = 2    (mod 3) = z_3.
So the eventual period is 3, which is the maximum possible over all c and z_0 when n = 3.

Maple

f:= proc(N)
  local pmax, R, S, T, z, a, b, c, x, y, found, zpd, count;
  pmax:= 0;
  for a from 0 to N-1 do
    for b from 0 to N-1 do
      c:= a+b*I;
      S:= {}:
      for x from 0 to N-1 do
        for y from 0 to N-1 do
           z:= x+I*y;
           if not member(z, S) then
             T:= {z};
             R[z]:= 0;
             found:= false;
             for count from 1 do
               z:= expand(z^2 + c) mod N;
               if member(z, S) then break fi;
               if member(z, T) then found:= true; zpd:= count - R[z]; break fi;
               R[z]:= count;
               T:= T union {z};
             od;
             S:= S union T;
             if found and zpd > pmax then
                pmax:= zpd fi;
           fi;
      od od;
  od od;
  pmax
end proc:
map(f, [$1..30]); # Robert Israel, Aug 29 2016

Mathematica

f[n_] := Module[{pmax = 0, R, S, T, z, a, b, c, x, y, found, zpd, count},
 For[a = 0, a <= n - 1, a++,
   For[b = 0, b <= n - 1, b++, c = a + b*I; S = {};
     For[x = 0, x <= n - 1, x++,
       For[y = 0, y <= n - 1, y++, z = x + y*I;
         If[!MemberQ[S, z], T = {z}; R[z] = 0; found = False;
         For[count = 1, True, count++,
           z = Mod[Expand[z^2 + c], n];
           If[MemberQ[S, z], Break[] ];
           If [MemberQ[T, z], found = True; zpd = count -
           R[z]; Break[]]; R[z] = count;
           T = Union[T, {z}]]; S = Union[S, T];
           If [found && zpd > pmax, pmax = zpd]]]]]];
  pmax];
Table[f[n], {n, 1, 20}] (* Jean-François Alcover, May 12 2023, after Robert Israel *)

Crossrefs

Sequence in context: A005421 A348083 A306156 * A208611 A209582 A158279

Adjacent sequences: A276122 A276123 A276124 * A276126 A276127 A276128

Keyword

nonn

Author

Luke Palmer, Aug 21 2016

Extensions

More terms from Robert Israel, Aug 29 2016

Status

approved