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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
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
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
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Nov 22 2002
STATUS
approved