OFFSET
1,2
COMMENTS
Sequence contains all powers of 5, infinitely many multiples of 12 and other numbers (including some factors of Fibonacci(5^j), e.g., 75025).
If m is in this sequence then 5*m is (since 5*m divides 5*F(m) which in turn divides F(5*m)). Also, if m is in this sequence then F(m) is in this sequence (since if gcd(F(m),m)=m then gcd(F(F(m)),F(m)) = F(gcd(F(m),m)) = F(m)). - Max Alekseyev, Sep 20 2009
From Max Alekseyev, Nov 29 2010: (Start)
Every term greater than 1 is a multiple of 5 or 12.
Proof. Let n>1 divide Fibonacci number F(n) and let p be the smallest prime divisor of n.
If p=2, then 3|n implying further that 4|n. Hence, 12|n.
If p=5, then 5|n.
If p is different from 2 and 5, then p divides either F(p+1) or F(p-1) and thus p divides either F(gcd(n,p+1)) or F(gcd(n,p-1)). Minimality of p implies that gcd(n,p-1)=1 and gcd(n,p+1)=1 (notice that p+1 being prime implies p=2 which is not the case). Therefore, p divides F(1)=1, a contradiction to the existence of such p. (End)
Luca & Tron give an upper bound, see links. - Charles R Greathouse IV, Aug 04 2021
REFERENCES
S. Wolfram, "A new kind of science", p. 891
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 500 terms from T. D. Noe, next 4600 terms from Lars Blomberg)
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 75.
Tamas Lengyel, Divisibility Properties by Multisection, Dec 2000.
Florian Luca and Emanuele Tron, The Distribution of Self-Fibonacci Divisors, Proceedings of the Thirteenth Conference of the Canadian Number Theory Association (CNTA XIII), Ayşe Alaca, Şaban Alaca, and Kenneth Williams, ed. (2015), pp. 149-158. arXiv:1410.2489 [math.NT], 2014.
C. Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.
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:
select(n -> fmod(n, n)=0, [$1..2000]); # Robert Israel, May 10 2016
MATHEMATICA
a=0; b=1; c=1; Do[a=b; b=c; c=a+b; If[Mod[c, n]==0, Print[n]], {n, 3, 1500}]
Select[Range[1350], Mod[Fibonacci[ # ], # ]==0&] (* Harvey P. Dale *)
PROG
(Haskell)
import Data.List (elemIndices)
a023172 n = a023172_list !! (n-1)
a023172_list =
map (+ 1) $ elemIndices 0 $ zipWith mod (tail a000045_list) [1..]
-- Reinhard Zumkeller, Oct 13 2011
(PARI) is(n)=((Mod([1, 1; 1, 0], n))^n)[1, 2]==0 \\ Charles R Greathouse IV, Feb 03 2014
(Magma) [n: n in [1..2*10^3] | Fibonacci(n) mod n eq 0 ]; // Vincenzo Librandi, Sep 17 2015
CROSSREFS
Cf. A128974 (12n does not divide Fibonacci(12n)). - Zak Seidov, Jan 10 2016
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by Don Reble, Sep 07 2003
STATUS
approved