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

Author

David W. Wilson

Keywords

• nonn

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)
• Oisín Flynn-Connolly, On the divisibility of sums of Fibonacci numbers, arXiv:2504.09938 [math.NT], 2025. See p. 1.
• Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 75.
• Tamas Lengyel, Divisibility Properties by Multisection, Dec 2000. Fib Q. 41 (1) 72
• 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.
• Chris Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.

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

Programs

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

Magma

[n: n in [1..2*10^3] | Fibonacci(n) mod n eq 0 ]; // Vincenzo Librandi, Sep 17 2015

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 *)

PARI

is(n)=((Mod([1,1;1,0],n))^n)[1,2]==0 \\ Charles R Greathouse IV, Feb 03 2014

Extensions

Edited by Don Reble, Sep 07 2003