A023172 Self-Fibonacci numbers: numbers k that divide Fibonacci(k).

1, 5, 12, 24, 25, 36, 48, 60, 72, 96, 108, 120, 125, 144, 168, 180, 192, 216, 240, 288, 300, 324, 336, 360, 384, 432, 480, 504, 540, 552, 576, 600, 612, 625, 648, 660, 672, 684, 720, 768, 840, 864, 900, 960, 972, 1008, 1080, 1104, 1152, 1176, 1200, 1224, 1296, 1320

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. A000350. See A127787 for an essentially identical sequence.

Cf. A000045, A069104, A123976, A159051, A263112.

Cf. A128974 (12n does not divide Fibonacci(12n)). - Zak Seidov, Jan 10 2016

Sequence in context: A130624 A344846 A066869 * A270681 A212540 A344510

Adjacent sequences: A023169 A023170 A023171 * A023173 A023174 A023175

Keyword

nonn

Author

David W. Wilson

Extensions

Edited by Don Reble, Sep 07 2003

Status

approved