A069104 -id:A069104 - cp's OEIS frontend

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

David W. Wilson

nonn

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

S. Wolfram, "A new kind of science", p. 891

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.

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

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

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

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

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 *)

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

Edited by Don Reble, Sep 07 2003

A069107 Composite numbers k that divide Fibonacci(k+1).

Original entry on oeis.org

323, 377, 2834, 3827, 5777, 6479, 10877, 11663, 18407, 19043, 20999, 23407, 25877, 27323, 34943, 35207, 39203, 44099, 47519, 50183, 51983, 53663, 60377, 65471, 75077, 78089, 79547, 80189, 81719, 82983, 84279, 84419, 86063, 90287, 94667

Offset: 1

Benoit Cloitre, Apr 06 2002

Primes p congruent to +2 or -2 (mod 5) divide Fibonacci(p+1) (cf. A003631 and [Hardy and Wright]).

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Fifth edition), Oxford Univ. Press (Clarendon), 1979, Chap. X, p. 150.

Chai Wah Wu, Table of n, a(n) for n = 1..2000 (n = 1..250 from Reinhard Zumkeller, n = 251..1000 from Giovanni Resta)

Cf. A045468, A003631, A064739, A081264 (Fibonacci pseudoprimes).

Cf. A000045, A010051, A023172, A069104.

a069107 n = a069107_list !! (n-1)
a069107_list = h 2 $ drop 3 a000045_list where
   h n (fib:fibs) = if fib `mod` n > 0 || a010051 n == 1
       then h (n+1) fibs else n : h (n+1) fibs
-- Reinhard Zumkeller, Oct 13 2011

Select[Range[2,100000],!PrimeQ[#]&&Divisible[Fibonacci[#+1],#]&] (* Harvey P. Dale, Sep 18 2011 *)

is(n)=((Mod([1,1;1,0],n))^(n+1))[1,2]==0 && !isprime(n) && n>1 \\ Charles R Greathouse IV, Oct 07 2016

Fibonacci(2*a(n)) mod a(n) = a(n) - 1. - Gary Detlefs, May 26 2014

Corrected by Ralf Stephan, Oct 17 2002

A123976 Numbers k such that Fibonacci(k-1) is divisible by k.

Original entry on oeis.org

1, 11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149, 151, 179, 181, 191, 199, 211, 229, 239, 241, 251, 269, 271, 281, 311, 331, 349, 359, 379, 389, 401, 409, 419, 421, 431, 439, 442, 449, 461, 479, 491, 499, 509, 521, 541, 569, 571, 599, 601

Offset: 1

Tanya Khovanova, Oct 30 2006

nonn

a(n) is a union of {1}, A069106(n) and A045468(n). Composite a(n) are listed in A069106(n) = {442, 1891, 2737, 4181, 6601, 6721, 8149, ...}. Prime a(n) are listed in A045468(n) = {11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149, 151, 179, 181, 191, 199, ...} Primes congruent to {1, 4} mod 5. - Alexander Adamchuk, Nov 02 2006

			Fibonacci(10) = 55, is divisible by 11.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A069106, A045468, A069104, A069107, A003631, A000045, A023172, A159051.

import Data.List (elemIndices)
a123976 n = a123976_list !! (n-1)
a123976_list = map (+ 1) $ elemIndices 0 $ zipWith mod a000045_list [1..]
-- Reinhard Zumkeller, Oct 13 2011

Select[Range[1000], IntegerQ[Fibonacci[ # - 1]/# ] &]

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

A159051 Numbers n such that Fibonacci(n-2) is divisible by n.

Original entry on oeis.org

2, 8, 38, 62, 122, 158, 218, 278, 302, 362, 398, 422, 458, 482, 542, 662, 698, 758, 818, 842, 878, 884, 902, 998, 1037, 1082, 1142, 1157, 1202, 1238, 1262, 1322, 1382, 1418, 1478, 1502, 1538, 1622, 1658, 1718, 1838, 1982, 2018, 2042, 2078, 2102, 2138

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 03 2009

nonn

			8=8/8=1, 38=14930352/38=392904, 62=1548008755920/62=24967883160, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A123976, A000045, A023172, A069104.

import Data.List (elemIndices)
a159051 n = a159051_list !! (n-1)
a159051_list = map (+ 2) $ elemIndices 0 $ zipWith mod a000045_list [2..]
-- Reinhard Zumkeller, Oct 13 2011

lst={};Do[If[Mod[(Fibonacci[n-2]),n]==0,AppendTo[lst,n]],{n,7!}];lst
Select[Range[2,2200],Divisible[Fibonacci[#-2],#]&] (* Harvey P. Dale, Dec 20 2014 *)

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

A100993 Indices k of Fibonacci numbers F(k) (A000045) that are divisible by k-1.

Original entry on oeis.org

2, 3, 4, 8, 14, 18, 24, 38, 44, 48, 54, 68, 74, 84, 98, 104, 108, 114, 128, 138, 158, 164, 168, 174, 194, 198, 224, 228, 234, 258, 264, 278, 284, 294, 308, 314, 318, 324, 338, 348, 354, 368, 374, 378, 384, 398, 434, 444, 458, 464, 468, 488, 504, 524, 548, 558

Offset: 1

Ron Knott, Nov 25 2004

nonn

When k-1 is prime, it is in A003631; when k-1 is composite (such as 323), it is in A069107. - T. D. Noe, Dec 13 2004

			14 is a term because F(14) = 377 = 13*29 is divisible by 13, one less than its index number 14.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ron Knott, Mathematical Magic of the Fibonacci Numbers.

Cf. A000045, A100992, A003631, A023162, A023173, A069104, A069107.

Select[Range[2, 500], Mod[ Fibonacci[ # ], # - 1] == 0 &] (* Robert G. Wilson v, Nov 26 2004 *)

a(n) = A069104(n) + 1. - T. D. Noe, Dec 13 2004

A220168 Numbers k that divide Fibonacci(k+2).

Original entry on oeis.org

1, 4, 34, 46, 88, 94, 106, 166, 214, 226, 274, 334, 346, 394, 454, 466, 514, 526, 586, 634, 646, 694, 706, 754, 766, 886, 934, 1006, 1114, 1126, 1174, 1186, 1234, 1294, 1306, 1354, 1366, 1486, 1546, 1594, 1654, 1714, 1726, 1774, 1894, 1906, 1954, 1966, 2026

Offset: 1

Alex Ratushnyak, May 03 2013

nonn

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A000045.
Cf. A023172 (numbers k that divide Fibonacci(k)).
Cf. A069104 (numbers k that divide Fibonacci(k+1)).
Cf. A123976 (numbers k that divide Fibonacci(k-1)).
Cf. A159051 (numbers k that divide Fibonacci(k-2)).

Select[Range[2000], Mod[Fibonacci[#+2], #] == 0 &] (* T. D. Noe, Feb 05 2014 *)

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

prpr = prev = 1
for i in range(3, 3000):
    prpr, prev = prev, prpr+prev
    if prev % (i-2) == 0:  print(i-2, end=', ')

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

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A069107 Composite numbers k that divide Fibonacci(k+1).

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A123976 Numbers k such that Fibonacci(k-1) is divisible by k.

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A159051 Numbers n such that Fibonacci(n-2) is divisible by n.

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A100993 Indices k of Fibonacci numbers F(k) (A000045) that are divisible by k-1.

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A220168 Numbers k that divide Fibonacci(k+2).

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