A023172 -id:A023172 - cp's OEIS frontend

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

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

A298685 Numbers i such that Fibonacci(i) is divisible by i, i+1, i+2, and i+3.

Original entry on oeis.org

540, 1200, 1620, 3060, 5580, 9180, 9900, 12600, 13440, 13680, 18300, 19440, 19800, 21000, 24480, 36900, 43200, 49680, 50220, 54120, 57240, 61560, 65880, 81180, 83700, 103680, 104160, 154080, 155520, 156060, 156240, 202440, 229320, 252000, 279000, 298200, 302940

Offset: 1

Alex Ratushnyak, Jan 24 2018

nonn

A subsequence of A298684.

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

Cf. A000045, A023172, A217738, A221018, A225219, A298684.

Select[Range[10^5], Function[{i, j}, AllTrue[i + Range[0, 3], Divisible[j, #] &]] @@ {#, Fibonacci@ #} &] (* Michael De Vlieger, Jan 28 2018 *)

isone(n, k) = !(fibonacci(n) % (n+k));
isok(n) = isone(n,0) && isone(n,1) && isone(n,2) && isone(n,3); \\ Michel Marcus, Jan 29 2018

A298686 Numbers i such that Fibonacci(i) is divisible by i+k for k=0,1,2,3,4.

Original entry on oeis.org

13440, 19440, 19800, 24480, 49680, 61560, 104160, 229320, 298200, 311040, 329400, 436800, 471240, 600600, 1202040, 1299600, 1468800, 1564920, 1702800, 2031120, 2352240, 2402400, 2499840, 2762760, 2805600, 2937600, 2962080, 3150840, 3262680, 3405600, 3843840

Offset: 1

Alex Ratushnyak, Jan 24 2018

nonn

A subsequence of A298685.

Chai Wah Wu, Table of n, a(n) for n = 1..147

Cf. A000045, A023172, A217738, A221018, A225219, A298684, A298685.

isone(n, k) = !(fibonacci(n) % (n+k));
isok(n) = isone(n,0) && isone(n,1) && isone(n,2) && isone(n,3) && isone(n,4); \\ Michel Marcus, Jan 28 2018

More terms from Alois P. Heinz, Jan 25 2018

A263112 a(n) = F(F(n)) mod n, where F = Fibonacci = A000045.

Original entry on oeis.org

0, 1, 1, 2, 0, 3, 2, 2, 1, 5, 1, 0, 8, 13, 10, 2, 12, 15, 5, 10, 1, 1, 1, 0, 0, 25, 1, 2, 5, 15, 27, 2, 10, 33, 20, 0, 1, 1, 34, 10, 40, 21, 18, 2, 10, 1, 1, 0, 1, 25, 1, 2, 16, 21, 5, 26, 37, 1, 7, 0, 33, 27, 1, 2, 40, 21, 5, 2, 1, 15, 1, 0, 46, 1, 25, 2, 68

Offset: 1

Alois P. Heinz, Oct 09 2015

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000045, A002708, A007570, A023172 (where a(n)=0), A263101, A274996, A338736.

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), n)[1, 2]:
seq(a(n), n=1..80);

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], n][[1, 2]];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Oct 29 2024, after Alois P. Heinz *)

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

A298687 Numbers i such that Fibonacci(i) is divisible by i+k for k=0..5.

Original entry on oeis.org

13440, 19440, 329400, 600600, 2499840, 3150840, 5590200, 7660800, 69069000, 83980800, 96049800, 98385840, 175472640, 179663400, 237484800, 320498640, 330663600, 375396840, 404351640, 406380240, 429660000, 437940000, 505234800, 574585200, 635980800

Offset: 1

Alex Ratushnyak, Jan 24 2018

nonn

A subsequence of A298686.

Chai Wah Wu, Table of n, a(n) for n = 1..57

Cf. A000045, A023172, A217738, A221018, A225219, A298684, A298685, A298686.

p0 = 0
p1 = 1
for i in range(1,1000000):
  if p1 % i == 0 and p1 % (i+1) == 0 and p1 % (i+2) == 0:
     if p1 % (i+3) == 0 and p1 % (i+4) == 0 and p1 % (i+5) == 0:  print(i)
  p0, p1 = p1, p0+p1

a(9)-a(25) from Chai Wah Wu, Jan 27 2018

A127787 Numbers n such that F(n) divides F(F(n)), where F(n) is a Fibonacci number.

Original entry on oeis.org

1, 2, 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

Alexander Adamchuk, May 13 2007

nonn

It is known that for n > 2 Fibonacci(n) divides Fibonacci(m) if and only if n divides m. Therefore if the term "2" is omitted this is identical to A023172, which see for further information. - Stefan Steinerberger, Dec 20 2007

			12 is a term because F(12) = 144 divides F(F(12)) = F(144) = 555565404224292694404015791808.

Cf. A023172. Cf. also A000045 = Fibonacci(n), A007570 = F(F(n)), where F is a Fibonacci number, A023172 = numbers n such that n divides Fibonacci(n).

Cf. A263101.

with(combinat): a:=proc(n) if type(fibonacci(fibonacci(n))/fibonacci(n), integer) then n else end if end proc: seq(a(n),n=1..40); # Emeric Deutsch, Aug 24 2007

Edited by N. J. A. Sloane, Dec 22 2007

A159231 Primes p such that 8p^2-2p-1 divides Fibonacci(p).

Original entry on oeis.org

37, 97, 577, 727, 1297, 3037, 3067, 4447, 4567, 5557, 7507, 7867, 8647, 9067, 9157, 12967, 17257, 20107, 20407, 21787, 22147, 23677, 25447, 27817, 28687, 29347, 30187, 32587, 33487, 35617, 38377, 42157, 42667, 42967, 43207, 45697, 46447, 47497, 49477

Offset: 1

Arkadiusz Wesolowski, Apr 06 2009

nonn

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
Chris Caldwell, The Prime Glossary, Fibonacci number
C. K. Caldwell, "Top Twenty" page, Fibonacci cofactor

Subsequence of A159259. Supersequence of A215158.

Cf. A000045, A023172, A159234, A181890.

[p : p in PrimesUpTo(49477) | IsZero(Fibonacci(p) mod (8*p^2-2*p-1))]; // Arkadiusz Wesolowski, Nov 09 2013

Select[Prime@Range[5084], Mod[Fibonacci[#], 8*#^2 - 2*# - 1] == 0 &] (* Arkadiusz Wesolowski, Dec 12 2011 *)

forprime(p=2, 49477, if(Mod(fibonacci(p), 8*p^2-2*p-1)==0, print1(p, ", "))); \\ Arkadiusz Wesolowski, Nov 09 2013

A159259 Positive numbers n such that 8n^2-2n-1 divides Fibonacci(n).

Original entry on oeis.org

27, 37, 97, 577, 687, 727, 777, 807, 1297, 1707, 1917, 2067, 2487, 2787, 2977, 3027, 3037, 3067, 3277, 3367, 3417, 3507, 3837, 4047, 4257, 4377, 4447, 4567, 4717, 5137, 5557, 5637, 5677, 5917, 5967, 6057, 6187, 6327, 7077, 7087, 7357, 7407, 7507, 7597

Offset: 1

Arkadiusz Wesolowski, Apr 07 2009

nonn

The prime numbers of this sequence are in A159231.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
Chris Caldwell, The Prime Glossary, Fibonacci number

Cf. A000045, A023172, A159234, A159231, A181890.

[n : n in [1..7597] | IsZero(Fibonacci(n) mod (8*n^2-2*n-1))] // Arkadiusz Wesolowski, Nov 09 2013

Select[Range[7597], Mod[Fibonacci[#], 8*#^2 - 2*# - 1] == 0 &] (* Arkadiusz Wesolowski, Dec 12 2011 *)

for(n=1, 7597, if(Mod(fibonacci(n), 8*n^2-2*n-1)==0, print1(n, ", "))); \\ Arkadiusz Wesolowski, Nov 09 2013

A219612 Numbers k that divide the sum of the first k Fibonacci numbers (beginning with F(0)).

Original entry on oeis.org

1, 4, 6, 9, 11, 19, 24, 29, 31, 34, 41, 46, 48, 59, 61, 71, 72, 79, 89, 94, 96, 100, 101, 106, 109, 120, 129, 131, 139, 144, 149, 151, 166, 179, 181, 191, 192, 199, 201, 211, 214, 216, 220, 226, 229, 239, 240, 241, 249, 251, 269, 271, 274, 281, 288, 311

Offset: 1

Alex Ratushnyak, May 03 2013

Numbers k such that A000045(k+1) == 1 (mod k). - Robert Israel, Oct 13 2015

			Sum of first 6 Fibonacci numbers is 0+1+1+2+3+5 = 12. Because 6 divides 12, 6 is in the sequence.

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

Cf. A000045, A000071, A023172, A045345, A101907.

fmod:= proc(a, b) local A, n, f1, f2, f;
  uses LinearAlgebra[Modular];
  A:= Mod(b, <<1, 1>|<1, 0>>, integer[8]);
  MatrixPower(b, M, a)[1, 2];
end proc:
1, op(select(t -> fmod(t+1,t) = 1, [$2..10^4])); # Robert Israel, Oct 13 2015

okQ[n_] := n == 1 || Mod[Fibonacci[n+1], n] == 1;
Select[Range[1000], okQ] (* Jean-François Alcover, Feb 04 2023 *)

lista(nn) = {sf = 0; for (n=0, nn, sf += fibonacci(n); if (sf % (n+1) == 0, print1(n+1, ", ")););} \\ Michel Marcus, Jun 05 2013

sum_, prpr, prev = 0, 0, 1
for i in range(1, 1000):
  sum_ += prpr
  if sum_ % i == 0:  print(i, end=', ')
  prpr, prev = prev, prpr+prev

a(n) = A101907(n) + 1. - Altug Alkan, Dec 29 2015

A352747 Array read by ascending antidiagonals. A(n, k) = F(k, n) mod n for n >= 1 and k >= 0, where F(n, k) = A352744(n, k) are the Fibonacci numbers, A(0, k) = 1 for k >= 0.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 1, 0, 1, 3, 1, 2, 0, 0, 1, 5, 3, 0, 1, 1, 0, 1, 1, 1, 3, 3, 0, 0, 0, 1, 5, 0, 3, 3, 2, 2, 1, 0, 1, 3, 2, 6, 5, 3, 1, 1, 0, 0, 1, 4, 1, 7, 5, 1, 3, 0, 0, 1, 0, 1, 0, 9, 8, 4, 4, 3, 3, 3, 2, 0, 0, 1, 5, 1, 4, 6, 1, 3, 5, 3, 2, 1, 1, 0, 1

Offset: 0

Peter Luschny, Apr 08 2022

This array aims the study of the divisibility properties of the Fibonacci numbers A352744. The identity F(n, k) = (-1)^k*F(1 - n, -k) from A352744 shows that negative indices do not add to the divisibility properties of F(n, k).
All rows A(n, .) are pure periodic sequences. The length of the periods is given by (1, A270313). For n > 0 the length of the period of row A(n, .) is <= n.
The period length is 1 for n in (1, A023172) and n for n in (1, A074215), as observed by Robert Israel in A270313. In particular, if n is a power of 2 or a prime (A174090), then the period length is n.
The indices of the zero-free rows are in A353280. A zero-free row A(n, .) means that n will not divide F(k, n) whatever value k takes. For that it is sufficient to check that period(A(n, .)) is zero-free.
If period(A(n, .)) = [k | 0 <= k < n] we call n a 'Fibonacci friend'. In other words, in this case F(k, n) mod n = k for 0 <= k < n. A Fibonacci friend does not have to be prime (since 1 is a Fibonacci friend), but if it is  prime then it is congruent to {1, 4} mod 5 (A045468), and all such primes are Fibonacci friends.
To say that n is a Fibonacci friend is equivalent to saying that A(n, n) = 0 and that n divides F(n, n). Fibonacci friends are the indices of the zeros in A002752.
Integers n > 0 that divide Sum{k=0..n-1} (F(k, n) mod n) are congruent to {0, 1, 3, 5} mod 6 (A301729).

			Array starts (periods are indicated with () ):
[n\k] 0   1   2   3   4  5  6   7   8   9  10  11  12
----------------------------------------------------------
[ 0] (1), 1,  1,  1,  1, 1, 1,  1,  1,  1,  1,  1,  1, ...
[ 1] (0), 0,  0,  0,  0, 0, 0,  0,  0,  0,  0,  0,  0, ...
[ 2] (1,  0), 1,  0,  1, 0, 1,  0,  1,  0,  1,  0,  1, ...
[ 3] (1,  0,  2), 1,  0, 2, 1,  0,  2,  1,  0,  2,  1, ...
[ 4] (2,  1,  0,  3), 2, 1, 0,  3,  2,  1,  0,  3,  2, ...
[ 5] (3), 3,  3,  3,  3, 3, 3,  3,  3,  3,  3,  3,  3, ...
[ 6] (5,  1,  3), 5,  1, 3, 5,  1,  3,  5,  1,  3,  5, ...
[ 7] (1,  0,  6,  5,  4, 3, 2), 1,  0,  6,  5,  4,  3, ...
[ 8] (5,  2,  7,  4,  1, 6, 3,  0), 5,  2,  7,  4,  1, ...
[ 9] (3,  1,  8,  6,  4, 2, 0,  7,  5), 3,  1,  8,  6, ...
[10] (4,  9), 4,  9,  4, 9, 4,  9,  4,  9,  4,  9,  4, ...
[11] (0,  1,  2,  3,  4, 5, 6,  7,  8,  9, 10), 0,  1, ...
[12] (5), 5,  5,  5,  5, 5, 5,  5,  5,  5,  5,  5,  5, ...

Cf. A352744, A002752, A045468, A074215, A088209, A174090, A212804, A270313, A301729, A353280.

f := n -> combinat:-fibonacci(n + 1):
F := proc(n, k) option remember; (n-1)*f(k-1) + f(k) end:
A := (n, k) -> ifelse(n = 0, 1, modp(F(k, n), n)):
for n from 0 to 12 do seq(A(n, k), k = 0..10) od;

F[n_, k_] := (n - 1)*Fibonacci[k] + Fibonacci[k + 1];
A[n_, k_] := If[n == 0, 1, Mod[F[k, n], n]];
Table[A[n, k], {n, 0, 12}, {k, 0, 10}] // TableForm

def F(n, k): return (n - 1)*fibonacci(k) + fibonacci(k + 1)
def A(n,k): return mod(F(k, n), n)
for n in range(13): print([A(n,k) for k in range(13)])

A(n, 0) = A(n, n) = A002752(n).

Clearly 0 <= A(n, k) < n for all k and n > 0.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

A298685 Numbers i such that Fibonacci(i) is divisible by i, i+1, i+2, and i+3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A298686 Numbers i such that Fibonacci(i) is divisible by i+k for k=0,1,2,3,4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A263112 a(n) = F(F(n)) mod n, where F = Fibonacci = A000045.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A298687 Numbers i such that Fibonacci(i) is divisible by i+k for k=0..5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Extensions

A127787 Numbers n such that F(n) divides F(F(n)), where F(n) is a Fibonacci number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Extensions

A159231 Primes p such that 8*p^2-2*p-1 divides Fibonacci(p).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A159259 Positive numbers n such that 8*n^2-2*n-1 divides Fibonacci(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

A159231 Primes p such that 8p^2-2p-1 divides Fibonacci(p).

A159259 Positive numbers n such that 8n^2-2n-1 divides Fibonacci(n).