A023173 -id:A023173 - cp's OEIS frontend

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

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

A132634 a(n) = Fibonacci(n) mod n^2.

Original entry on oeis.org

0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 0, 64, 181, 160, 219, 152, 316, 210, 365, 362, 287, 91, 288, 25, 389, 317, 291, 378, 440, 869, 261, 574, 339, 765, 432, 443, 533, 1285, 1355, 1641, 1504, 85, 1741, 20, 551, 1832, 576, 1457, 1525, 389, 803, 2066, 332, 1820, 245

Offset: 1

Hieronymus Fischer, Aug 24 2007

nonn

a(n)=0 for n=1 and n=12 only (conjecture).

			a(13) = 64, since Fibonacci(13) = 233 == 64 (mod 13^2).

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from Hieronymus Fischer)

Cf. A000045, A023173, A236395.

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

Table[Mod[Fibonacci[n],n^2],{n,200}] (* Vladimir Joseph Stephan Orlovsky, Nov 28 2010 *)

A121343 a(n) = Fibonacci(n) mod n(n+1)/2.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 8, 13, 21, 34, 0, 23, 66, 51, 62, 10, 35, 67, 19, 1, 45, 89, 1, 229, 168, 275, 298, 236, 319, 59, 155, 125, 309, 376, 407, 485, 630, 628, 419, 466, 615, 370, 517, 343, 663, 830, 988, 1033, 168, 624, 700, 746, 1167, 158, 872, 1105, 609, 610, 59, 1181, 0, 1, 125

Offset: 0

N. J. A. Sloane, Aug 29 2006

			a(11)=23 since Fib(11)=89==23(mod (11*12/2)).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000045, A023173, A000217, A096535.

a:= proc(n) local r, M, p, m; r, M, p, m:=
      <<1|0>, <0|1>>, <<0|1>, <1|1>>, n, n*(n+1)/2;
      do if irem(p, 2, 'p')=1 then r:= r.M mod m fi;
         if p=0 then break fi; M:= M.M mod m
      od; r[1, 2]
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 26 2016

f[n_] := If[n == 0, 0, Mod[Fibonacci@n, n(n + 1)/2]]; f /@ Range[0, 62] (* Robert G. Wilson v, Aug 31 2006 *)
Join[{0},Mod[First[#],Last[#]]&/@With[{nn=70},Thread[{Fibonacci[ Range[ nn]], Accumulate[Range[nn]]}]]] (* Harvey P. Dale, May 21 2012 *)

fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
a(n)=lift(fibmod(n,n*(n+1)/2)) \\ Charles R Greathouse IV, Jun 20 2017

A000045(n) modulo A000217(n).

Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar

A132636 a(n) = Fibonacci(n) mod n^3.

Original entry on oeis.org

0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 1685, 7063, 4323, 4896, 12525, 15937, 19271, 10483, 2060, 22040, 5674, 15621, 2752, 3807, 9340, 432, 46989, 19305, 11932, 62155, 31899, 12088, 22273, 3677, 32420

Offset: 1

Hieronymus Fischer, Aug 24 2007

nonn

a(1) = 0; for n=2 to 20, a(n) = A000045(n); otherwise for instance, see example. - Michel Marcus, Jul 15 2013

			a(21) = 1685, since Fibonacci(21) = 10946 == 1685 (mod 21^3).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000045, A000578, A023173.

Table[Mod[Fibonacci[n],n^3],{n,45}] (* James C. McMahon, Mar 08 2025 *)

a(n) = fibonacci(n) % n^3 \\ Michel Marcus, Jul 15 2013

a(n)=lift((Mod([1,1;1,0],n^3))^n)[1,2] \\ Charles R Greathouse IV, Jul 18 2013

a(n) = A000045(n) mod A000578(n).

A100992 Indices k of Fibonacci numbers F(k) (A000045) that are divisible by k+1.

Original entry on oeis.org

10, 18, 28, 30, 40, 58, 60, 70, 78, 88, 100, 108, 130, 138, 148, 150, 178, 180, 190, 198, 210, 228, 238, 240, 250, 268, 270, 280, 310, 330, 348, 358, 378, 388, 400, 408, 418, 420, 430, 438, 441, 448, 460, 478, 490, 498, 508, 520, 540, 568, 570, 598, 600, 618

Offset: 1

Ron Knott, Nov 25 2004

nonn

When k+1 is prime, it is in A045468; when k+1 is composite (such as 442), it is in A069106. - T. D. Noe, Dec 13 2004

			18 is a term because F(18) = 2584 = 2*2*2*17*19 is divisible by 19, one more than its index number 18.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ron Knott, Mathematical Magic of the Fibonacci Numbers.

Cf. A000045, A100993, A023162, A023173, A045468, A069106.

Select[ Range[ 620], Mod[ Fibonacci[ # ], # + 1] == 0 &] (* Robert G. Wilson v, Nov 26 2004 *)

More terms from Robert G. Wilson v, Nov 26 2004

A100993 Indices k of Fibonacci numbers F(k) (A000045) that are divisible by k-1.

Original entry on oeis.org

2, 3, 4, 8, 14, 18, 24, 38, 44, 48, 54, 68, 74, 84, 98, 104, 108, 114, 128, 138, 158, 164, 168, 174, 194, 198, 224, 228, 234, 258, 264, 278, 284, 294, 308, 314, 318, 324, 338, 348, 354, 368, 374, 378, 384, 398, 434, 444, 458, 464, 468, 488, 504, 524, 548, 558

Offset: 1

Ron Knott, Nov 25 2004

nonn

When k-1 is prime, it is in A003631; when k-1 is composite (such as 323), it is in A069107. - T. D. Noe, Dec 13 2004

			14 is a term because F(14) = 377 = 13*29 is divisible by 13, one less than its index number 14.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ron Knott, Mathematical Magic of the Fibonacci Numbers.

Cf. A000045, A100992, A003631, A023162, A023173, A069104, A069107.

Select[Range[2, 500], Mod[ Fibonacci[ # ], # - 1] == 0 &] (* Robert G. Wilson v, Nov 26 2004 *)

a(n) = A069104(n) + 1. - T. D. Noe, Dec 13 2004

A128288 a(n) = A023163(n)/3 for n > 1.

Original entry on oeis.org

3, 13, 37, 43, 53, 67, 83, 107, 157, 163, 173, 197, 227, 277, 283, 293, 307, 317, 347, 373, 397, 443, 467, 523, 547, 557, 563, 587, 613, 643, 653, 677, 683, 733, 757, 773, 787, 797, 827, 853, 877, 883, 907, 947, 997, 1013, 1093, 1117, 1123, 1163, 1187, 1213

Offset: 2

Alexander Adamchuk, Feb 24 2007

nonn

3 divides A023163(n) for n > 1. A023163(n) are the numbers n such that Fibonacci(n) == -2 (mod n). Almost all terms of {a(n)} are prime that belong to A003631 = {2, 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97} (primes congruent to {2, 3} mod 5) that are also the primes p that divide Fibonacci(p+1). The first composite term is a(74) = 1853 = 17*109. The second composite term is 9701 = 89*109. The third composite term is 10877 = 73*149 belong to A069107(n) Composite n such that n divides F(n+1) where F(k) are the Fibonacci numbers. Composite terms in {a(n)} are listed in A128289 = {1853, 9701, 10877, 17261, ...}.

			A023163 begins {1, 9, 39, 111, 129, 159, 201, 249, 321, 471, 489, 519, ...}.
Thus a(2) = A023163(2)/3 = 9/3 = 3, a(3) = A023163(3)/3 = 39/3 = 13.

Cf. A002708, A023172, A023173, A023162, A023163 (numbers k such that Fibonacci(k) == -2 (mod k)).

Cf. A003631, A069107, A128289 (composite terms in A128288).

a(n) = A023163(n)/3 for n > 1.

A319040 Numbers k > 1 such that Pell(k) == 1 (mod k).

Original entry on oeis.org

7, 17, 23, 31, 35, 41, 47, 71, 73, 79, 89, 97, 103, 113, 127, 137, 151, 167, 169, 191, 193, 199, 223, 233, 239, 241, 257, 263, 271, 281, 311, 313, 337, 353, 359, 367, 383, 385, 401, 409, 431, 433, 439, 449, 457, 463, 479, 487, 503, 521, 569, 577, 593, 599

Offset: 1

Jon E. Schoenfield, Sep 08 2018

nonn

It appears that most of the terms of this sequence are primes. The composite terms are 35, 169, 385, 899, 961, 1121, ... (A319042).

The primes in the sequence give A001132 (primes == +-1 (mod 8)), since for primes p we have Pell(p) == (2/p) (mod p) where (2/p) is the Legendre symbol. - Jianing Song, Sep 10 2018

			k = 7 is in the sequence since Pell(7) = 169 = 7 * 24 + 1 == 1 (mod 7).
k = 11 is not in the sequence: Pell(11) = 5741 = 11 * 522 - 1 !== 1 (mod 11).
k = 35 is in the sequence: Pell(35) = 8822750406821 = 35 * 252078583052 + 1 == 1 (mod 35).

Cf. A000129 (Pell numbers), A001132, A023173, A319041, A319042, A319043.

isA319040 := k -> simplify(2^(k-1)*hypergeom([1-k/2,(1-k)/2],[1-k],-1)) mod k = 1: A319040List := b -> select(isA319040, [$1..b]):
A319040List(600); # Peter Luschny, Sep 09 2018

Select[Range[500], Mod[Fibonacci[#, 2], #] == 1 &] (* Alonso del Arte, Sep 08 2018 *)

A128289 Composite terms in A128288(n) = A023163(n)/3 for n>1.

Original entry on oeis.org

1853, 9701, 10877, 17261, 23323, 27403, 75077, 80189, 113573, 120581, 161027, 162133, 163059, 196877, 213749, 291941, 361397, 400987, 427549, 482677, 635627, 667589, 941291, 1030373, 1033997, 1140701, 1196061, 1256293, 1751747, 1816363, 1842581, 2288453, 2662277

Offset: 1

Alexander Adamchuk, Feb 24 2007

nonn

3 divides A023163(n) for n>1. A023163(n) are the numbers n such that Fibonacci(n) == -2 (mod n).
Almost all terms of A128288 are prime that belong to A003631 = {2, 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97} Primes congruent to {2, 3} mod 5; that are also the primes p that divide Fibonacci(p+1).
a(3) = 10877 = 73*149 belongs to A069107 Composite n such that n divides Fibonacci(n+1).
a(3) = 10877 and a(4) = 17261 belong to A094395 Odd composite n such that n divides Fibonacci(n) + 1.

			a(1) = A128288(74) = 1853 = 17*109.
a(2) = 9701 = 89*109.
a(3) = 10877 = 73*149.
a(4) = 17261 = 41*421.
a(5) = 23323 = 83*281.

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

Cf. A128288, A002708, A023172, A023173, A023162, A023163 = numbers n such that Fib(n) == -2 (mod n). Cf. A003631, A069107, A094413, A094395 = Odd composite n such that n divides Fibonacci(n) + 1.

Do[ f = Mod[ Fibonacci[3n], 3n ]; If[ !PrimeQ[n] && f == 3n-2, Print[ {n, FactorInteger[n]} ]], {n,1,25000} ]

Two more terms from R. J. Mathar, Oct 08 2007

a(9)-a(33) from Amiram Eldar, Apr 07 2019

A023177 Numbers k such that Fibonacci(k) == 8 (mod k).

Original entry on oeis.org

1, 3, 6, 54, 174, 246, 366, 534, 606, 654, 894, 966, 1086, 1374, 1446, 1614, 1686, 2046, 2094, 2214, 2334, 2406, 2454, 2526, 2694, 2766, 3054, 3126, 3246, 3414, 3606, 3846, 3966, 4206, 4254, 4566, 4614, 4854, 4926, 4974, 5286, 5382, 5406, 5574, 5646, 6054

Offset: 1

David W. Wilson

nonn

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

Cf. A000045, A002708, A023165, A023173, A023174, A023175, A023176.

filter:= n -> combinat:-fibonacci(n) - 8 mod n = 0:
select(filter, [$1..10^4]); # Robert Israel, Mar 05 2025

Select[Range[6100], Mod[Fibonacci[#] - 8, #]==0&]  (* Harvey P. Dale, Sep 21 2021 *)

A002708(a(n)) = 8 (for n >= 4). - Robert Israel, Mar 05 2025

Definition clarified by N. J. A. Sloane, Sep 21 2021

cp's OEIS Frontend

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

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A132634 a(n) = Fibonacci(n) mod n^2.

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A121343 a(n) = Fibonacci(n) mod n(n+1)/2.

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A132636 a(n) = Fibonacci(n) mod n^3.

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A100992 Indices k of Fibonacci numbers F(k) (A000045) that are divisible by k+1.

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A100993 Indices k of Fibonacci numbers F(k) (A000045) that are divisible by k-1.

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A128288 a(n) = A023163(n)/3 for n > 1.

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