A127787 -id:A127787 - 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

A263101 a(n) = F(F(n)) mod F(n), where F = Fibonacci = A000045.

Original entry on oeis.org

0, 0, 1, 2, 0, 5, 12, 5, 33, 5, 1, 0, 232, 233, 55, 5, 1596, 2563, 1, 5, 987, 10946, 28656, 0, 0, 75025, 189653, 89, 1, 6765, 1, 5, 6765, 1, 9227460, 0, 24157816, 1, 63245985, 5, 1, 267914275, 433494436, 4181, 1134896405, 1, 2971215072, 0, 7778741816, 75025

Offset: 1

Alois P. Heinz, Oct 09 2015

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A000045, A002708, A007570, A076240, A127787 (where a(n)=0), A263112, A274996.

F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
p:= (M, n, k)-> map(x-> x mod k, `if`(n=0, <<1|0>, <0|1>>,
          `if`(n::even, p(M, n/2, k)^2, p(M, n-1, k).M))):
a:= n-> p(<<0|1>, <1|1>>, F(n)$2)[1, 2]:
seq(a(n), n=1..50);

F[n_] := MatrixPower[{{0, 1}, {1, 1}}, n][[1, 2]];
p[M_, n_, k_] := Mod[#, k]& /@ If[n == 0, {{1, 0}, {0, 1}}, If[EvenQ[n], MatrixPower[p[M, n/2, k], 2], p[M, n - 1, k].M]];
a[n_] := p[{{0, 1}, {1, 1}}, F[n], F[n]][[1, 2]];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 29 2024, after Alois P. Heinz *)

alist(nn)= my(f=fibonacci); [ f(f(n))%f(n) |n<-[1..nn] ]; \\ Ruud H.G. van Tol, Dec 13 2024

a(n) = A007570(n) mod A000045(n).

A270313 Denominator of Fibonacci(n)/n.

Original entry on oeis.org

1, 2, 3, 4, 1, 3, 7, 8, 9, 2, 11, 1, 13, 14, 3, 16, 17, 9, 19, 4, 21, 22, 23, 1, 1, 26, 27, 28, 29, 3, 31, 32, 33, 34, 7, 1, 37, 38, 39, 8, 41, 21, 43, 44, 9, 46, 47, 1, 49, 2, 51, 52, 53, 27, 11, 8, 57, 58, 59, 1, 61, 62, 63, 64, 13, 33, 67, 68, 69, 14, 71, 1, 73, 74, 3

Offset: 1

Jean-François Alcover and Paul Curtz, Mar 15 2016

a(n) = 1 for n in A023172; a(n) = n for n in A074215. - Robert Israel, Mar 16 2016

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000045, A023172, A074215, A104714, A127787, A270312 (numerators).

seq(n/igcd(n,combinat:-fibonacci(n)), n=1..100); # Robert Israel, Mar 16 2016

Table[Fibonacci[n]/n, {n, 1, 100}] // Denominator

a(n) = denominator(fibonacci(n)/n); \\ Michel Marcus, Mar 16 2016

a(n) = n/A104714(n). - Robert Israel, Mar 16 2016

A270312 Numerator of Fibonacci(n)/n.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 13, 21, 34, 11, 89, 12, 233, 377, 122, 987, 1597, 1292, 4181, 1353, 10946, 17711, 28657, 1932, 3001, 121393, 196418, 317811, 514229, 83204, 1346269, 2178309, 3524578, 5702887, 1845493, 414732, 24157817, 39088169, 63245986, 20466831, 165580141

Offset: 1

Jean-François Alcover and Paul Curtz, Mar 15 2016

The fractions are an autosequence of the second kind. See the link.
Array of fractions and successive differences:
   1,   1/2,   2/3,   3/4,      1, ...
-1/2,   1/6,  1/12,   1/4,    1/3, ...
2 /3, -1/12,   1/6,  1/12,   4/21, ...
-3/4,   1/4, -1/12,  3/28,   3/56, ...
   1,  -1/3,  4/21, -3/56, 11/126, ...
...
The sequence of fractions being an autosequence, it can be noticed that first column, which is the inverse binomial transform of first row, is identical to the sequence, up to alternating signs.
In addition, main diagonal is twice the first upper diagonal (autosequence of the second kind).

			Fractions begin:
1, 1/2, 2/3, 3/4, 1, 4/3, 13/7, 21/8, 34/9, 11/2, 89/11, 12, ...

OEIS Wiki, Autosequence

Cf. A000045, A023172, A127787, A270313 (denominators).

Table[Fibonacci[n]/n, {n, 1, 50}] // Numerator

a(n) = numerator(fibonacci(n)/n); \\ Michel Marcus, Mar 15 2016

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

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A263101 a(n) = F(F(n)) mod F(n), where F = Fibonacci = A000045.

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A270313 Denominator of Fibonacci(n)/n.

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A270312 Numerator of Fibonacci(n)/n.

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