A078345 Numbers k such that F(k) mod k divides F(F(k) mod k) 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

Author

Benoit Cloitre, Nov 22 2002

Keywords

• nonn

Examples

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

Links

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

Crossrefs

Cf. A000045, A002708, A023172.

Programs

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 *)

Formula

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