A256342 Moduli n for which A248218(n) = 2 (length of the terminating cycle of 0 under x -> x^2+1 modulo n).

2, 4, 6, 8, 11, 12, 14, 16, 22, 23, 24, 28, 29, 32, 33, 38, 42, 44, 46, 48, 53, 56, 58, 62, 64, 66, 67, 69, 74, 76, 77, 84, 86, 87, 88, 92, 96, 106, 107, 109, 112, 114, 116, 124, 127, 128, 132, 134, 138, 148, 152, 154, 159, 161, 163, 168, 172, 174, 176, 184, 186, 192

Offset: 1

Author

M. F. Hasler, Mar 25 2015

Keywords

• nonn

Comments

If x is a member and y is a member of this sequence or A248219, then LCM(x,y) is a member. - Robert Israel, Mar 09 2021

Examples

			In Z/mZ with m = 2, the iteration of x -> x^2+1 starting at x = 0 yields (0, 1, 0, ...), and m = 2 is the least positive number for which there is such a cycle of length 2, here [0, 1], therefore a(1) = 2.
For m = 3, the iteration yields (0, 1, 2, 2, ...), i.e., a cycle [2] of length 1, therefore 3 is not in this sequence.
For m = 4, the iterations yield (0, 1, 2, 1, ...), and since there is again a cycle [1, 2] of length 2, a(2)=4.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A248218, A248219, A256343 - A256349, A003095, A247981.

Programs

Maple

filter:= proc(n) local x, k, R,p;
  x:= 0; R[0]:= 0;
  for k from 1 do
    x:= x^2+1 mod n;
    if assigned(R[x]) then return evalb(k-R[x] = 2)
    else R[x]:= k
    fi
  od;
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 09 2021

Mathematica

filterQ[n_] := Module[{x, k, R}, x = 0; R[0] = 0; For[k = 1, True, k++, x = Mod[x^2 + 1, n]; If[IntegerQ[R[x]], Return[k - R[x] == 2], R[x] = k]]];
Select[Range[1000], filterQ] (* Jean-François Alcover, Feb 01 2023, after Robert Israel *)

PARI

for(i=1,200,A248218(i)==2&&print1(i","))