A023172 -id:A023172 - cp's OEIS frontend

A016089 Numbers n such that n divides n-th Lucas number A000032(n).

1, 6, 18, 54, 162, 486, 1458, 1926, 4374, 5778, 13122, 17334, 39366, 52002, 118098, 156006, 206082, 354294, 468018, 618246, 1062882, 1404054, 1854738, 2471058, 3188646, 4212162, 5564214, 7413174, 9565938, 12636486, 16692642, 22050774

Offset: 1

Robert G. Wilson v

nonn

Note that if n divides A000032(n) and p is an odd prime divisor of A000032(n), then pn divides A000032(pn) and, furthermore, p^k*n divides A000032(p^k*n) for every integer k>=0.
In particular, since 6 divides A000032(6) = 2*3^2, A016089 includes all terms of the geometric progression 2*3^k for k>0 (see A099856); since 18 divides A000032(18) = 2*3^3*107, A016089 includes all terms of the form 2*107^m*3^k for k>1 and m>=0; etc.
Terms of A016089 starting with 18 are multiples of 18. There are no other terms of the form 18p where p is prime, except for p=3 and p=107. - Alexander Adamchuk, May 11 2007

Lars Blomberg, Table of n, a(n) for n = 1..91
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 75.
C. Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.

Cf. A000032, A000204, A025192, A008776.

Cf. A099856, A072378 = numbers n such that 12n divides Fibonacci(12n), A023172 = numbers n such that n divides Fibonacci(n).

a = 1; b = 3; Do[c = a + b; a = b; b = c; If[Mod[c, n] == 0, Print[n]], {n, 3, 2, 10^6}]

is(n)=(Mod([0,1;1,1],n)^n*[2;1])[1,1]==0 \\ Charles R Greathouse IV, Nov 04 2016

Extended and revised by Max Alekseyev, May 13 2007, May 15 2008, May 16 2008

A002708 a(n) = Fibonacci(n) mod n.

Original entry on oeis.org

0, 1, 2, 3, 0, 2, 6, 5, 7, 5, 1, 0, 12, 13, 10, 11, 16, 10, 1, 5, 5, 1, 22, 0, 0, 25, 20, 11, 1, 20, 1, 5, 13, 33, 30, 0, 36, 1, 37, 35, 1, 34, 42, 25, 20, 45, 46, 0, 36, 25, 32, 23, 52, 8, 5, 21, 40, 1, 1, 0, 1, 1, 43, 59, 60, 52, 66, 65, 44, 15, 1, 0, 72, 73, 50, 3, 2, 44, 1, 5, 7, 1, 82, 24

Offset: 1

John C. Hallyburton, Jr. (hallyb(AT)evms.ENET.dec.com)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 891.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 5000 terms from T. D. Noe)
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)

Cf. A002726, A002752, A023172 (indices of 0's), A023173 (indices of 1's), A023174-A023182.
Cf. A263101.
Main diagonal of A161553.

[Fibonacci(n) mod n : n in [1..120]]; // Vincenzo Librandi, Nov 19 2015

with(combinat): [ seq( fibonacci(n) mod n, n=1..80) ];
# second Maple program:
a:= proc(n) local r, M, p; r, M, p:=
      <<1|0>, <0|1>>, <<0|1>, <1|1>>, n;
      do if irem(p, 2, 'p')=1 then r:= r.M mod n fi;
         if p=0 then break fi; M:= M.M mod n
      od; r[1, 2]
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 26 2016

Table[Mod[Fibonacci[n], n], {n, 1, 100}] (* Stefan Steinerberger, Apr 18 2006 *)

a(n) = fibonacci(n) % n; \\ Michel Marcus, May 11 2016

A002708_list, a, b, = [], 1, 1
for n in range(1,10**4+1):
    A002708_list.append(a%n)
    a, b = b, a+b # Chai Wah Wu, Nov 26 2015

More terms from Stefan Steinerberger, Apr 18 2006

A232570 Numbers k that divide tribonacci(k) (A000073(k)).

Original entry on oeis.org

1, 8, 16, 19, 32, 47, 53, 64, 103, 112, 128, 144, 155, 163, 192, 199, 208, 221, 224, 256, 257, 269, 272, 299, 311, 368, 397, 401, 419, 421, 448, 499, 512, 587, 599, 617, 640, 683, 757, 768, 773, 784, 863, 883, 896, 907, 911, 929, 936, 991, 1021, 1024

Offset: 1

Seiichi Manyama, Jun 17 2016

nonn

Inspired by A023172 (numbers k such that k divides Fibonacci(k)).
Includes all primes p such that x^3-x^2-x-1 has 3 distinct roots in the field GF(p) (A106279). - Robert Israel, Feb 07 2018
Includes 2^k for k >= 3. - Robert Israel, Jul 26 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000073, A023172, A106279.

with(LinearAlgebra[Modular]):
T:= (n, m)-> MatrixPower(m, Mod(m, <<0|1|0>,
    <0|0|1>, <1|1|1>>, float[8]), n)[1, 3]:
a:= proc(n) option remember; local k; if n=1
      then 1 else for k from 1+a(n-1)
      while T(k$2)>0 do od; k fi
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Feb 05 2018

trib = LinearRecurrence[{1, 1, 1}, {0, 0, 1}, 2000]; Reap[Do[If[Divisible[ trib[[n+1]], n], Print[n]; Sow[n]], {n, 1, Length[trib]-1}]][[2, 1]] (* Jean-François Alcover, Mar 22 2019 *)

require 'matrix'
def power(a, n, mod)
  return Matrix.I(a.row_size) if n == 0
  m = power(a, n >> 1, mod)
  m = (m * m).map{|i| i % mod}
  return m if n & 1 == 0
  (m * a).map{|i| i % mod}
end
def f(m, n)
  ary0 = Array.new(m, 0)
  ary0[0] = 1
  v = Vector.elements(ary0)
  ary1 = [Array.new(m, 1)]
  (0..m - 2).each{|i|
    ary2 = Array.new(m, 0)
    ary2[i] = 1
    ary1 << ary2
  }
  a = Matrix[*ary1]
  mod = n
  (power(a, n, mod) * v)[m - 1]
end
def a(n)
  (1..n).select{|i| f(3, i) == 0}
end

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

A104714 Greatest common divisor of a Fibonacci number and its index.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 2, 1, 1, 1, 5, 1, 12, 1, 1, 5, 1, 1, 2, 1, 5, 1, 1, 1, 24, 25, 1, 1, 1, 1, 10, 1, 1, 1, 1, 5, 36, 1, 1, 1, 5, 1, 2, 1, 1, 5, 1, 1, 48, 1, 25, 1, 1, 1, 2, 5, 7, 1, 1, 1, 60, 1, 1, 1, 1, 5, 2, 1, 1, 1, 5, 1, 72, 1, 1, 25, 1, 1, 2, 1, 5, 1, 1, 1, 12, 5, 1, 1, 1, 1, 10, 13, 1, 1, 1, 5, 96, 1

Offset: 0

Harmel Nestra (harmel.nestra(AT)ut.ee), Apr 23 2005

Considering this sequence is a natural sequel to the investigation of the problem when F_n is divisible by n (the numbers occurring in A023172). This sequence has several nice properties. (1) n | m implies a(n) | a(m) for arbitrary naturals n and m. This property is a direct consequence of the analogous well-known property of Fibonacci numbers. (2) gcd (a(n), a(m)) = a(gcd(n, m)) for arbitrary naturals n and m. Also this property follows directly from the analogous (perhaps not so well-known) property of Fibonacci numbers. (3) a(n) * a(m) | a(n * m) for arbitrary naturals n and m. This property is remarkable especially in the light that the analogous proposition for Fibonacci numbers fails if n and m are not relatively prime (e.g. F_3 * F_3 does not divide F_9). (4) The set of numbers satisfying a(n) = n is closed w.r.t. multiplication. This follows easily from (3).

			The natural numbers:    0 1 2 3 4 5 6  7  8  9 10 11  12 ...
The Fibonacci numbers:  0 1 1 2 3 5 8 13 21 34 55 89 144 ...
The corresponding GCDs: 0 1 1 1 1 5 2  1  1  1  5  1  12 ...

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 1001 terms from T. D. Noe)
Paolo Leonetti, Carlo Sanna, On the greatest common divisor of n and the nth Fibonacci number, arXiv:1704.00151 [math.NT], 2017.
Carlo Sanna, Emanuele Tron, The density of numbers n having a prescribed G.C.D. with the nth Fibonacci number, arXiv:1705.01805 [math.NT], 2017.

Cf. A023172, A000045, A001177, A001175, A001176. a(n) = gcd(A000045(n), A001477(n)). a(n) = n iff n occurs in A023172 iff n | A000045(n).

Cf. A074215 (a(n)==1).

let fibs@(_ : fs) = 0 : 1 : zipWith (+) fibs fs in 0 : zipWith gcd [1 ..] fs

b:= proc(n) option remember; local r, M, p; r, M, p:=
      <<1|0>, <0|1>>, <<0|1>, <1|1>>, n;
      do if irem(p, 2, 'p')=1 then r:= r.M mod n fi;
         if p=0 then break fi; M:= M.M mod n
      od; r[1, 2]
    end:
a:= n-> igcd(n, b(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 05 2017

Table[GCD[Fibonacci[n],n],{n,0,97}] (* Alonso del Arte, Nov 22 2010 *)

a(n)=if(n,gcd(n,lift(Mod([1,1;1,0],n)^n)[1,2]),0) \\ Charles R Greathouse IV, Sep 24 2013

a(n) = gcd(F(n), n).

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

A221018 Number n such that Fibonacci(n) is divisible by n, n + 1 and n - 1.

Original entry on oeis.org

108, 2688, 3528, 4968, 6480, 15288, 18048, 20808, 21000, 25308, 35448, 47520, 62928, 68208, 81648, 82008, 97608, 103968, 108288, 116928, 137088, 139968, 151848, 162288, 196560, 197568, 200928, 235008, 238728, 253368, 278208, 363888, 379008, 400248, 444528, 449568, 457608

Offset: 1

Alex Ratushnyak, May 03 2013

nonn

Numbers n such that Fibonacci(n) mod n = Fibonacci(n) mod (n-1) = Fibonacci(n) mod (n+1) = 0.

Subsequence of A023172.

Chai Wah Wu, Table of n, a(n) for n = 1..200

Cf. A000045, A023172, A225219.

prpr, prev = 0, 1
for n in range(2, 100000):
    prpr, prev = prev, prpr+prev
    if n%2: continue
    if (prev % n, prev % (n-1), prev % (n+1)) == (0, 0, 0):
        print(n, end=", ")

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

A298684 Numbers i such that Fibonacci(i) is divisible by i, i+1, and i+2.

Original entry on oeis.org

60, 540, 660, 1200, 1320, 1620, 2160, 3060, 5580, 6120, 6600, 6720, 8100, 9180, 9240, 9600, 9720, 9900, 11160, 12240, 12300, 12600, 13200, 13440, 13680, 15120, 15360, 18300, 18480, 19440, 19800, 21000, 22500, 24480, 24840, 26880, 27360, 28920, 29400, 30240, 30780

Offset: 1

Alex Ratushnyak, Jan 24 2018

nonn

A subsequence of A217738.

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A000045, A023172, A217738, A221018, A225219.

[n: n in [2..10^5] | IsZero(Fibonacci(n) mod (n)) and IsZero(Fibonacci(n) mod (n+1)) and IsZero(Fibonacci(n) mod (n+2))]; // Vincenzo Librandi, Jan 27 2018

fQ[n_] := Mod[ Fibonacci@ n, {n, n +1, n +2}] == {0, 0, 0}; Select[60 Range@513, fQ] (* Robert G. Wilson v, Jan 26 2018 *)

from _future_ import division
A298684_list, n, a, b = [], 1, 1, 1
while len(A298684_list) < 1000:
    if not (a % (n*(n+1)*(n+2)//(1 if n % 2 else 2))):
        A298684_list.append(n)
    n += 1
    a, b = b, a+b # Chai Wah Wu, Jan 26 2018

A000350 Numbers m such that Fibonacci(m) ends with m.

Original entry on oeis.org

0, 1, 5, 25, 29, 41, 49, 61, 65, 85, 89, 101, 125, 145, 149, 245, 265, 365, 385, 485, 505, 601, 605, 625, 649, 701, 725, 745, 749, 845, 865, 965, 985, 1105, 1205, 1249, 1345, 1445, 1585, 1685, 1825, 1925, 2065, 2165, 2305, 2405, 2501, 2545, 2645, 2785, 2885

Offset: 1

N. J. A. Sloane

Conjecture: Other than 1 and 5, there is no m such that Fibonacci(m) in binary ends with m in binary. The conjecture holds up to m=50000. - Ralf Stephan, Aug 21 2006
The conjecture for binary numbers holds for m < 2^25. - T. D. Noe, May 14 2007
Conjecture is true. It is easy to prove (by induction on k) that if Fibonacci(m) ends with m in binary, then m == 0, 1, or 5 (mod 3*2^k) for any positive integer k, i.e., m must simply be equal to 0, 1, or 5. - Max Alekseyev, Jul 03 2009

			Fibonacci(25) = 75025 ends with 25.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1034 (terms n = 1..803 from T. D. Noe)
G. R. Deily, Terminal Digit Coincidences Between Fibonacci Numbers and Their Indices, The Fibonacci Quarterly, 4.2 (1966) 151.
M. Dunton and R. E. Grimm, Fibonacci on Egyptian fractions, Fib. Quart., 4 (1966), 339-354.
D. A. Lind, Extended Computations of Terminal Digit Coincidences, Fibonacci Quarterly, 5.2 April 1967 pp. 183-184.

Cf. A000045, A050816, A038546, A052000, A023172.

import Data.List (isSuffixOf, elemIndices)
import Data.Function (on)
a000350 n = a000350_list !! (n-1)
a000350_list = elemIndices True $
               zipWith (isSuffixOf `on` show) [0..] a000045_list
-- Reinhard Zumkeller, Apr 10 2012

a=0;b=1;c=1;lst={}; Do[a=b;b=c;c=a+b;m=Floor[N[Log[10,n]]]+1; If[Mod[c,10^m]==n,AppendTo[lst,n]],{n,3,5000}]; Join[{0,1},lst] (* edited and changed by Harvey P. Dale, Sep 10 2011 *)
fnQ[n_]:=Mod[Fibonacci[n],10^IntegerLength[n]]==n; Select[Range[ 0,2900],fnQ] (* Harvey P. Dale, Nov 03 2012 *)

for(n=0,1e4,if(((Mod([1,1;1,0],10^#Str(n)))^n)[1,2]==n,print1(n", "))) \\ Charles R Greathouse IV, Apr 10 2012

from sympy import fibonacci
[i for i in range(1000) if str(fibonacci(i))[-len(str(i)):]==str(i)] # Nicholas Stefan Georgescu, Feb 27 2023

cp's OEIS Frontend

A016089 Numbers n such that n divides n-th Lucas number A000032(n).

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A002708 a(n) = Fibonacci(n) mod n.

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A232570 Numbers k that divide tribonacci(k) (A000073(k)).

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

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A104714 Greatest common divisor of a Fibonacci number and its index.

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

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A221018 Number n such that Fibonacci(n) is divisible by n, n + 1 and n - 1.

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