A159051 -id:A159051 - cp's OEIS frontend

A159052 Odd terms in A159051.

1037, 1157, 20737, 250973, 854813, 1055617, 4042469, 8588477

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 03 2009

nonn

lst={};Do[If[Mod[(Fibonacci[n-2]),n]==0,If[OddQ[n],AppendTo[lst,n]]],{n,3*8!}];lst
a = c = 0; b = 1; lst = {}; Do[c = a + b; If[ Mod[a, n] == 0 && OddQ@n, Print@n]; a = b; b = c, {n, 2, 10^7}] (* Robert G. Wilson v, Apr 05 2009 *)

a(4) - a(8) from Robert G. Wilson v, Apr 05 2009
Edited by N. J. A. Sloane, Apr 06 2009
Erroneous term 1 deleted by N. J. A. Sloane, Dec 20 2014

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

Original entry on oeis.org

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

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

A069104 Numbers m such that m divides Fibonacci(m+1).

Original entry on oeis.org

1, 2, 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 103, 107, 113, 127, 137, 157, 163, 167, 173, 193, 197, 223, 227, 233, 257, 263, 277, 283, 293, 307, 313, 317, 323, 337, 347, 353, 367, 373, 377, 383, 397, 433, 443, 457, 463, 467, 487, 503, 523, 547, 557

Offset: 1

Benoit Cloitre, Apr 06 2002

Equals A003631 union A069107.

Let u(1)=u(2)=1 and (m+2)*u(m+2) = (m+1)*u(m+1) + m*u(m); then sequence gives values of k such that u(k) is an integer.

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

Cf. A000045, A023172, A123976, A159051.

import Data.List (elemIndices)
a069104 n = a069104_list !! (n-1)
a069104_list =
   map (+ 1) $ elemIndices 0 $ zipWith mod (drop 2 a000045_list) [1..]
-- Reinhard Zumkeller, Oct 13 2011

Select[Range[6! ],IntegerQ[Fibonacci[ #+1]/# ]&] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2009 *)
Select[Range[600],Mod[Fibonacci[#+1],#]==0&] (* Harvey P. Dale, Feb 24 2025 *)

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

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=', ')

A159053 Indices n such that Fibonacci(n-3) is a multiple of n.

Original entry on oeis.org

1, 3, 87, 123, 143, 183, 267, 303, 327, 447, 483, 543, 687, 723, 807, 843, 1023, 1047, 1167, 1203, 1227, 1263, 1347, 1383, 1527, 1563, 1623, 1707, 1763, 1803, 1923, 1983, 2103, 2127, 2283, 2307, 2427, 2463, 2487, 2643, 2691, 2703, 2787, 2823, 3027, 3063

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 03 2009

nonn

			Fibonacci(84)/87=160500643816367088/87=1844834986395024, so 87 is in the sequence. Fibonacci(120)/123=5358359254990966640871840/123 = 43563896382040379194080 so 123 is in the sequence.

Cf. A123976, A159051, A159052.

lst={};Do[If[Mod[(Fibonacci[n-3]),n]==0,AppendTo[lst,n]],{n,3*7!}];lst
Select[Range[3100],Mod[Fibonacci[#-3],#]==0&] (* Harvey P. Dale, Jul 18 2024 *)

Comments edited by R. J. Mathar, Apr 05 2009

A159054 Members of A159053 which are not multiples of 3.

Original entry on oeis.org

1, 143, 1763, 5183, 10153, 10403, 15109, 36863, 40183, 79523, 97343, 130273

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 03 2009

nonn

Cf. A123976, A159051, A159052, A159053.

lst={};Do[If[Mod[(Fibonacci[n-3]),n]==0,If[Mod[n,3]!=0,AppendTo[lst,n]]], {n,5*8!}];lst

A159053 INTERSECT A001651. - R. J. Mathar, Apr 05 2009

Definition reworded by R. J. Mathar, Apr 05 2009

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A159052 Odd terms in A159051.

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

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

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A069104 Numbers m such that m divides Fibonacci(m+1).

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

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A159053 Indices n such that Fibonacci(n-3) is a multiple of n.

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A159054 Members of A159053 which are not multiples of 3.

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