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

%I A078345 #18 Jun 27 2025 17:50:30
%S A078345 1,2,5,8,10,11,12,13,19,20,21,22,24,25,26,29,31,32,36,37,38,41,44,48,
%T A078345 49,50,55,58,59,60,61,62,65,71,72,73,79,80,82,84,89,95,96,97,101,104,
%U A078345 108,109,118,120,122,125,131,132,139,140,142,144,145,149,151,155,156
%N A078345 Numbers k such that F(k) mod k divides F(F(k) mod k) where F(k) denotes the k-th Fibonacci number.
%H A078345 Robert Israel, <a href="/A078345/b078345.txt">Table of n, a(n) for n = 1..10000</a>
%F A078345 Conjecture: a(n) is asymptotic to c*n*log(n) with c>0.7
%e A078345 F(44) = 701408733; 701408733 mod 44 = 25, F(25)=75025 and 25 divides 75025, hence 44 is in the sequence.
%p A078345 fmod:= proc(n,m) local M,t; uses LinearAlgebra:-Modular;
%p A078345     if m <= 1 then return 0 fi;
%p A078345     if m < 2^25 then t:= float[8] else t:= integer fi;
%p A078345     M:= Mod(m,<<1,1>|<1,0>>,t);
%p A078345     round(MatrixPower(m,M,n)[1,2])
%p A078345 end proc:
%p A078345 filter:= proc(n) local s;
%p A078345   s:= fmod(n,n);
%p A078345   fmod(s,s) = 0
%p A078345 end proc:
%p A078345 select(filter, [$1..200]); # _Robert Israel_, May 10 2016
%t A078345 Unprotect[Divisible];
%t A078345 Divisible[0, 0] = True;
%t A078345 okQ[n_] := Module[{F = Fibonacci, m}, m = Mod[F[n], n];  Divisible[F[m], m]];
%t A078345 Select[Range[75000], okQ] (* _Jean-François Alcover_, Jul 09 2024 *)
%Y A078345 Cf. A000045, A002708, A023172.
%K A078345 nonn
%O A078345 1,2
%A A078345 _Benoit Cloitre_, Nov 22 2002