A123976 -id:A123976 - 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

A045468 Primes congruent to {1, 4} mod 5.

Original entry on oeis.org

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, 449, 461, 479, 491

Offset: 1

N. J. A. Sloane

Rational primes that decompose in the field Q(sqrt(5)). - N. J. A. Sloane, Dec 26 2017
These are also primes p that divide Fibonacci(p-1). - Jud McCranie
Primes ending in 1 or 9. - Lekraj Beedassy, Oct 27 2003
Also primes p such that p divides 5^(p-1)/2 - 4^(p-1)/2. - Cino Hilliard, Sep 06 2004
Primes p such that the polynomial x^2-x-1 mod p has 2 distinct zeros. - T. D. Noe, May 02 2005
Same as A038872, apart from the term 5. - R. J. Mathar, Oct 18 2008
Appears to be the primes p such that p^6 mod 210 = 1. - Gary Detlefs, Dec 29 2011
Primes in A047209, also in A090771. - Reinhard Zumkeller, Jan 07 2012
Primes p such that p does not divide Sum_{i=1..p} Fibonacci(i)^2. The sum is A001654(p). - Arkadiusz Wesolowski, Jul 23 2012
Primes congruent to {1, 9} mod 10. Legendre symbol (5, a(n)) = +1. For prime 5 this symbol (5, 5) is set to 0, and (5, prime) = -1 for prime == {3, 7} (mod 10), given in A003631. - Wolfdieter Lang, Mar 05 2021

Hardy and Wright, An Introduction to the Theory of Numbers, Chap. X, p. 150, Oxford University Press, Fifth edition.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Caleb Ji, Tanya Khovanova, Robin Park, and Angela Song, Chocolate Numbers, arXiv:1509.06093 [math.CO], 2015.
Caleb Ji, Tanya Khovanova, Robin Park, and Angela Song, Chocolate Numbers, Journal of Integer Sequences, Vol. 19 (2016), #16.1.7.
Index to sequences related to decomposition of primes in quadratic fields

Cf. A030430, A030433, A064739, A038872, A010051, A003631.

Subsequence of A123976.

a045468 n = a045468_list !! (n-1)
a045468_list = [x | x <- a047209_list, a010051 x == 1]
-- Reinhard Zumkeller, Jan 07 2012

[ p: p in PrimesUpTo(1000) | p mod 5 in {1,4} ]; // Vincenzo Librandi, Aug 13 2012

for n from 1 to 500 do if(isprime(n)) and (n^6 mod 210=1) then print(n) fi od;  # Gary Detlefs, Dec 29 2011

lst={};Do[p=Prime[n];If[Mod[p,5]==1||Mod[p,5]==4,AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)
Select[Prime[Range[200]],MemberQ[{1,4},Mod[#,5]]&] (* Vincenzo Librandi, Aug 13 2012 *)

list(lim)=select(n->n%5==1||n%5==4,primes(primepi(lim))) \\ Charles R Greathouse IV, Jul 25 2011

A069106 Composite numbers k such that k divides F(k-1) where F(j) are the Fibonacci numbers.

Original entry on oeis.org

442, 1891, 2737, 4181, 6601, 6721, 8149, 13201, 13981, 15251, 17119, 17711, 30889, 34561, 40501, 51841, 52701, 64079, 64681, 67861, 68101, 68251, 78409, 88601, 88831, 90061, 96049, 97921, 115231, 118441, 138601, 145351, 146611, 150121, 153781, 163081, 179697, 186961, 191351, 194833

Offset: 1

Benoit Cloitre, Apr 06 2002

Primes p congruent to 1 or 4 (mod 5) divide F(p-1) (cf. A045468 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.

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

Subsequence of A123976.
Cf. A045468, A003631, A064739, A081264 (Fibonacci pseudoprimes).
Cf. A002808, A000045.

#include  #include  #define STARTN 10 #define N_OF_MILLER_RABIN_TESTS 5 int main() { mpz_t n, f1, f2; int flag=0; /* flag? 0: f1 contains current F[n-1] 1: f2 = F[n-1] */ mpz_set_ui (n, STARTN); mpz_init (f1); mpz_init (f2); mpz_fib2_ui (f1, f2, STARTN-1); for (;;) { if (mpz_probab_prime_p (n, N_OF_MILLER_RABIN_TESTS)) goto next_iter; if (mpz_divisible_p (!flag? f1:f2, n)) { mpz_out_str (stdout, 10, n); printf (" "); fflush (stdout); } next_iter: mpz_add_ui (n, n, 1); mpz_add (!flag? f2:f1, f1, f2); flag = !flag; } }

a069106 n = a069106_list !! (n-1)
a069106_list = [x | x <- a002808_list, a000045 (x-1) `mod` x == 0]
-- Reinhard Zumkeller, Jul 19 2013

A069106[nn_] := Select[Complement[Range[2,nn],Prime[Range[2,PrimePi[ nn]]]],Divisible[ Fibonacci[ #-1],#]&] (* Harvey P. Dale, Jul 05 2011 *)

fibmod(n,m)=((Mod([1,1;1,0],m))^n)[1,2]
is(n)=!isprime(n) && !fibmod(n-1,n) && n>1 \\ Charles R Greathouse IV, Oct 06 2016

Corrected and extended (with C program) by Ralf Stephan, Oct 13 2002

a(35)-a(40) added by Reinhard Zumkeller, Jul 19 2013

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

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

A159052 Odd terms in A159051.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 03 2009

nonn

Cf. A123976, A159051.

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

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

cp's OEIS Frontend

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

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A045468 Primes congruent to {1, 4} mod 5.

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A069106 Composite numbers k such that k divides F(k-1) where F(j) are the Fibonacci numbers.

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

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

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A159052 Odd terms in A159051.

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

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