A263101 -id:A263101 - 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

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.

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

A274996 a(n) = F(F(F(n))) mod F(F(n)), where F = Fibonacci = A000045.

Original entry on oeis.org

0, 0, 0, 1, 0, 5, 232, 987, 1, 5, 1, 0, 2211236406303914545699412969744873993387956988652, 2211236406303914545699412969744873993387956988653, 139583862445

Offset: 1

Alois P. Heinz, Nov 11 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..19

Cf. A000045, A007570, A058051, A263101, A263112, A338889.

F:= 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:=
        `if`(nargs=1, r.M, r.M mod args[2]) fi;
         if p=0 then break fi; M:=
        `if`(nargs=1, M.M, M.M mod args[2])
      od; r[1, 2]
    end:
a:= n-> (h-> F(h$2))(F(F(n))):
seq(a(n), n=1..15);

a(n) = A058051(n) mod A007570(n).

A338638 a(n) = L(L(n)) mod L(n), where L = Lucas numbers = A000032.

Original entry on oeis.org

1, 0, 1, 3, 1, 1, 0, 1, 1, 7, 4, 1, 199, 1, 4, 843, 1, 1, 0, 1, 29, 123, 4, 1, 3, 199, 4, 39603, 29, 1, 5778, 1, 1, 7, 4, 17622890, 12752043, 1, 4, 39603, 7881196, 1, 5778, 1, 29, 7, 4, 1, 3, 1149851, 28143689044, 7, 29, 1, 0, 312119004790, 6643838879, 7, 4, 1

Offset: 0

Alois P. Heinz, Nov 04 2020

Alois P. Heinz, Table of n, a(n) for n = 0..4795

Cf. A000032, A005371, A016089, A213060, A263101, A338736, A338889.

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:=
        `if`(nargs=1, r.M, r.M mod args[2]) fi;
         if p=0 then break fi; M:=
        `if`(nargs=1, M.M, M.M mod args[2])
      od; (r.<<2, 1>>)[1$2]
    end:
a:= n-> (f-> b(f$2) mod f)(b(n)):
seq(a(n), n=0..60);

Table[Mod[LucasL[LucasL[n]],LucasL[n]],{n,0,60}] (* Harvey P. Dale, Jul 04 2022 *)

a(n) = A005371(n) mod A000032(n).

a(n) = 0 for n in { A016089 }.

cp's OEIS Frontend

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

A274996 a(n) = F(F(F(n))) mod F(F(n)), where F = Fibonacci = A000045.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A338638 a(n) = L(L(n)) mod L(n), where L = Lucas numbers = A000032.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula