A002708 -id:A002708 - cp's OEIS frontend

A182167 Min( f(n), n-f(n) ), where f(n) = A002708(n) = Fibonacci(n) mod n.

0, 1, 1, 1, 0, 2, 1, 3, 2, 5, 1, 0, 1, 1, 5, 5, 1, 8, 1, 5, 5, 1, 1, 0, 0, 1, 7, 11, 1, 10, 1, 5, 13, 1, 5, 0, 1, 1, 2, 5, 1, 8, 1, 19, 20, 1, 1, 0, 13, 25, 19, 23, 1, 8, 5, 21, 17, 1, 1, 0, 1, 1, 20, 5, 5, 14, 1, 3, 25, 15, 1, 0, 1, 1, 25, 3, 2, 34, 1, 5, 7

Offset: 1

Alex Ratushnyak, Apr 15 2012

nonn

Conjecture: the most frequent values are 0,1,2,3,5,8,13,21,34,... i.e Fibonacci numbers.

			a(1) = min( A002708(1) , 1 - A002708(1) ) = min(0,1) = 0,  a(4) = min(3,1) = 1,  a(5) = min(0,5) = 0

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

Cf. A000045, A002708.

a(n)=my(f=lift(((Mod([1,1;1,0],n))^n)[1,2]));min(f,n-f) \\ Charles R Greathouse IV, Apr 16 2012

a(n) = min( A002708(n) , n - A002708(n) )

a(n) = min( Fibonacci(n) mod n , n - (Fibonacci(n) mod n) )

A161553 Table which contains in row n the fundamental Pisano period of the Fibonacci sequence (mod n).

Original entry on oeis.org

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

Offset: 1

Alexander Adamchuk, Jun 13 2009

The length of the n-th row (the length of the period) is A001175(n).

			F(n) mod 1 {0},
F(n) mod 2 {0,1,1},
F(n) mod 3 {0,1,1,2,0,2,2,1},
F(n) mod 4 {0,1,1,2,3,1},
F(n) mod 5 {0,1,1,2,3,0,3,3,1,4,0,4,4,3,2,0,2,2,4,1},
F(n) mod 6 {0,1,1,2,3,5,2,1,3,4,1,5,0,5,5,4,3,1,4,5,3,2,5,1},
F(n) mod 7 {0,1,1,2,3,5,1,6,0,6,6,5,4,2,6,1},
F(n) mod 8 {0,1,1,2,3,5,0,5,5,2,7,1},
F(n) mod 9 {0,1,1,2,3,5,8,4,3,7,1,8,0,8,8,7,6,4,1,5,6,2,8,1},
F(n) mod 10 {0,1,1,2,3,5,8,3,1,4,5,9,4,3,7,0,7,7,4,1,5,6,1,7,8,5,3,8, 1,9,0,9,9,8,7,5,2,7,9,6,5,1,6,7,3,0,3,3,6,9,5,4,9,3,2,5,7,2,9,1}.

Alois P. Heinz, Rows n = 1..200, flattened
J. D. Fulton and W. L. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arithm. 16 (1969) 106-110.
Wayne Peng, ABC Implies There are Infinitely Many non-Fibonacci-Wieferich Primes - An Application of ABC Conjecture over Number Fields, arXiv:1511.05645 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Pisano Period.
Wikipedia, Pisano period.

Cf. A000045, A001175.
Main diagonal gives A002708.
Row sums give A214300.

per[1] = 1; per[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0 && Mod[Fibonacci[k + 1], n] == 1, Return[k]]];
row[n_] := Table[Mod[Fibonacci[k], n], {k, 0, per[n]-1}];
Array[row, 9] // Flatten (* Jean-François Alcover, Oct 30 2018 *)

row(n)={my(L=List([0]), X=Mod([1,1;1,0],n), I=Mod([1,0;0,1],n), M=X); while(M<>I, M*=X; listput(L, lift(M[2,2]))); Vec(L)} \\ Andrew Howroyd, Mar 05 2023

Moved into the keyword:tabf category by R. J. Mathar, Oct 04 2009

A263101 a(n) = F(F(n)) mod F(n), where F = Fibonacci = A000045.

Original entry on oeis.org

0, 0, 1, 2, 0, 5, 12, 5, 33, 5, 1, 0, 232, 233, 55, 5, 1596, 2563, 1, 5, 987, 10946, 28656, 0, 0, 75025, 189653, 89, 1, 6765, 1, 5, 6765, 1, 9227460, 0, 24157816, 1, 63245985, 5, 1, 267914275, 433494436, 4181, 1134896405, 1, 2971215072, 0, 7778741816, 75025

Offset: 1

Alois P. Heinz, Oct 09 2015

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A000045, A002708, A007570, A076240, A127787 (where a(n)=0), A263112, A274996.

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)$2)[1, 2]:
seq(a(n), n=1..50);

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], F[n]][[1, 2]];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 29 2024, after Alois P. Heinz *)

alist(nn)= my(f=fibonacci); [ f(f(n))%f(n) |n<-[1..nn] ]; \\ Ruud H.G. van Tol, Dec 13 2024

a(n) = A007570(n) mod A000045(n).

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.

A002752 a(n) = Fibonacci(n-1) mod n.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 1, 5, 3, 4, 0, 5, 1, 9, 2, 2, 1, 13, 0, 1, 3, 12, 1, 1, 18, 15, 1, 26, 0, 29, 0, 29, 12, 2, 22, 17, 1, 1, 29, 26, 0, 13, 1, 13, 33, 2, 1, 1, 21, 49, 37, 18, 1, 23, 47, 13, 39, 30, 0, 41, 0, 1, 62, 34, 8, 49, 1, 5, 3, 54, 0, 1, 1, 39, 7, 2, 74

Offset: 1

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

nonn

T. D. Noe, Table of n, a(n) for n=1..5000

Column 0 of A352747.

Cf. A000045, A002708,

Table[Mod[Fibonacci[n],n+1],{n,6!}] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2010 *)

a(n) = fibonacci(n-1) % n; \\ Michel Marcus, Apr 11 2022

def A002752(n): return mod(fibonacci(n - 1), n)
# Assuming offset 0 this prepends a(0) = 1.
print([A002752(n) for n in range(79)]) # Peter Luschny, Apr 11 2022

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.

A213060 Lucas(n) mod n, Lucas(n)= A000032(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 0, 1, 7, 4, 3, 1, 10, 1, 3, 14, 15, 1, 0, 1, 7, 11, 3, 1, 2, 11, 3, 22, 7, 1, 18, 1, 31, 4, 3, 4, 34, 1, 3, 17, 7, 1, 18, 1, 7, 41, 3, 1, 2, 29, 23, 4, 7, 1, 0, 44, 47, 4, 3, 1, 22, 1, 3, 41, 63, 11, 18, 1, 7, 50, 53, 1, 2, 1, 3, 64, 7, 73, 18

Offset: 1

Gary Detlefs, Jun 03 2012

a(n) = 1 for all prime values of n. Composite values for which a(n) = 1 are listed in A005845.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000032, A005845.

Cf. A002708 (Fibonacci(n) mod n).

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

with(combinat):f:=n-> fibonacci(n):L:=n->f(2*n)/f(n): seq(L(n) mod n, n= 1..75)
# alternative
A213060 := proc(n::integer)
    modp(A000032(n),n) ;
end proc:
seq(A213060(n),n=1..100) ; # R. J. Mathar, Oct 02 2019

Table[Mod[LucasL[n], n], {n, 100}] (* T. D. Noe, Jun 06 2012 *)

A002726 a(n) = Fibonacci(n+1) mod n.

Original entry on oeis.org

0, 0, 0, 1, 3, 1, 0, 2, 1, 9, 1, 5, 0, 8, 12, 13, 0, 5, 1, 6, 8, 13, 0, 1, 18, 14, 21, 9, 1, 19, 1, 2, 25, 1, 17, 17, 0, 2, 27, 21, 1, 5, 0, 38, 8, 1, 0, 1, 8, 24, 18, 41, 0, 31, 52, 34, 22, 31, 1, 41, 1, 2, 42, 29, 3, 35, 0, 2, 47, 69, 1, 1, 0, 38, 57, 5, 76, 31, 1, 66, 55, 43, 0, 5, 3, 44

Offset: 1

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

nonn

T. D. Noe, Table of n, a(n) for n=1..5000

Cf. A000045, A002708.

Table[Mod[Fibonacci[n+1],n],{n,6!}] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2010 *)

PARI
```
a(n)=fibonacci(n+1)%n
```

More terms from Michael Somos, Apr 26 2000

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

A182625 Numbers n for which Fibonacci(n) mod n is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 14, 19, 20, 21, 22, 24, 25, 29, 31, 32, 33, 36, 38, 41, 42, 48, 54, 55, 56, 58, 59, 60, 61, 62, 71, 72, 76, 77, 79, 80, 82, 89, 92, 93, 95, 96, 101, 104, 105, 108, 109, 110, 118, 119, 120, 121, 122, 123, 124, 125, 131, 133, 139, 142

Offset: 1

Carmine Suriano, Mar 30 2011

			Fibonacci(12) = 144, 144 mod 12 = 0, and 0 is a Fibonacci number. Therefore 12 is in the sequence.
Fibonacci(14) = 377, 377 mod 14 = 13, and 13 is a Fibonacci number. Therefore 14 is in the sequence.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 2241 terms from Klaus Brockhaus)

Cf. A000045, A002708.

isA000045 := proc(n) local F,i; for i from 0 do F := combinat[fibonacci](i) ; if F> n then return false; elif F = n then return true; end if; end do;end proc:
isA182625 := proc(n) isA000045(combinat[fibonacci](n) mod n) ; end proc:
for n from 1 to 300 do if isA182625(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Apr 02 2011
# second Maple program:
b:= 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:
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1)) while (t->
         not (issqr(t+4) or issqr(t-4)))(5*b(k)^2)
      do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 26 2016

nn=12; f=Table[Fibonacci[n], {n,0,nn}]; Select[Range[f[[-1]]], MemberQ[f, Mod[Fibonacci[#],#]]&] (* T. D. Noe, Apr 02 2011 *)

is(n)=my(k=(fibonacci(n)%n)^2);k+=(k+1)<<2; issquare(k) || issquare(k-8) \\ Charles R Greathouse IV, Jul 30 2012

{n: A002708(n) in A000045}. - R. J. Mathar, Apr 02 2011

cp's OEIS Frontend

A182167 Min( f(n), n-f(n) ), where f(n) = A002708(n) = Fibonacci(n) mod n.

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A161553 Table which contains in row n the fundamental Pisano period of the Fibonacci sequence (mod n).

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A263101 a(n) = F(F(n)) mod F(n), where F = Fibonacci = A000045.

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

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A002752 a(n) = Fibonacci(n-1) mod n.

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Sage

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

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A213060 Lucas(n) mod n, Lucas(n)= A000032(n).

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A002726 a(n) = Fibonacci(n+1) mod n.

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