A078345 Numbers n such that F(n) mod n divides F(F(n) mod n) where F(k) denotes the k-th Fibonacci number.

1, 2, 5, 8, 10, 11, 12, 13, 19, 20, 21, 22, 24, 25, 26, 29, 31, 32, 36, 37, 38, 41, 44, 48, 49, 50, 55, 58, 59, 60, 61, 62, 65, 71, 72, 73, 79, 80, 82, 84, 89, 95, 96, 97, 101, 104, 108, 109, 118, 120, 122, 125, 131, 132, 139, 140, 142, 144, 145, 149, 151, 155, 156

Offset

1,2

Links

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

Formula

Conjecture: a(n) is asymptotic to c*n*log(n) with c>0.7

Example

F(44) = 701408733; 701408733 mod 44 = 25, F(25)=75025 and 25 divides 75025, hence 44 is in the sequence.

Maple

fmod:= proc(n, m) local M, t; uses LinearAlgebra:-Modular;
    if m <= 1 then return 0 fi;
    if m < 2^25 then t:= float[8] else t:= integer fi;
    M:= Mod(m, <<1, 1>|<1, 0>>, t);
    round(MatrixPower(m, M, n)[1, 2])
end proc:
filter:= proc(n) local s;
  s:= fmod(n, n);
  fmod(s, s) = 0
end proc:
select(filter, [$1..200]); # Robert Israel, May 10 2016

Mathematica

Unprotect[Divisible];
Divisible[0, 0] = True;
okQ[n_] := Module[{F = Fibonacci, m}, m = Mod[F[n], n];  Divisible[F[m], m]];
Select[Range[75000], okQ] (* Jean-François Alcover, Jul 09 2024 *)

Crossrefs

Cf. A000045, A002708, A023172.

Sequence in context: A083724 A072476 A167541 * A161817 A080228 A153052

Adjacent sequences: A078342 A078343 A078344 * A078346 A078347 A078348

Keyword

nonn

Author

Benoit Cloitre, Nov 22 2002

Status

approved