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

1,1

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

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A195873 A337854 A073140 * A171820 A193879 A226722

Adjacent sequences: A256339 A256340 A256341 * A256343 A256344 A256345

Keyword

nonn

Author

M. F. Hasler, Mar 25 2015

Status

approved