cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-20 of 20 results.

A217737 a(n) = Fibonacci(n) mod n*(n+1).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 51, 167, 130, 171, 67, 190, 1, 45, 320, 1, 505, 168, 275, 649, 614, 319, 59, 620, 125, 837, 376, 407, 485, 1296, 1331, 419, 466, 1435, 1231, 1420, 1289, 1653, 830, 2069, 2161, 1344, 1849, 1975, 746, 1167, 1589, 872, 2645, 2205
Offset: 1

Views

Author

Alex Ratushnyak, Mar 22 2013

Keywords

Links

Crossrefs

Cf. A000045, A002708, A121343, A132634, A132636.

Programs

  • Maple
    a:= proc(n) local r, M, p, m; r, M, p, m:=
      <<1|0>, <0|1>>, <<0|1>, <1|1>>, n, n*(n+1);
      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=1..100);  # Alois P. Heinz, Nov 26 2016
  • Mathematica
    Table[Mod[Fibonacci[n],n(n+1)],{n,60}] (* Harvey P. Dale, Oct 02 2017 *)
  • PARI
    a(n)=fibonacci(n)%(n*(n+1)) \\ Charles R Greathouse IV, Jun 23 2017
  • Python
    prpr, prev = 0, 1
for i in range(1, 333):
    cur = prpr + prev
    print(str(prev % (i*(i+1))), end=', ')
    prpr, prev = prev, cur

Formula

A000045(n) modulo A002378(n).

A023164 Numbers k such that Fibonacci(k) == -3 (mod k).

Original entry on oeis.org

1, 2, 8, 68, 92, 188, 212, 332, 428, 452, 548, 668, 692, 788, 908, 932, 1028, 1052, 1172, 1268, 1292, 1388, 1412, 1508, 1532, 1772, 1868, 2012, 2074, 2156, 2228, 2252, 2314, 2348, 2372, 2468, 2588, 2612, 2708, 2732, 2972, 3092, 3188, 3308, 3428, 3452, 3548
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Includes 4*p for primes p with p == 17 or 23 (mod 30). - Robert Israel, May 11 2021

Links

Crossrefs

Cf. A000045, A002708.

Programs

  • Maple
    fpp:= n -> mpow(n-1, n)[2, 2]:
M:= <<0, 1>|<1, 1>>:
mpow:= proc(n, p)
  if n = 0 then <<1, 0>|<0, 1>>
  elif n::even then procname(n/2, p)^2 mod p
  else  procname((n-1)/2, p)^2 . M mod p
  fi
end proc:
select(t -> fpp(t)+3 mod t = 0, [$1..10000]); # Robert Israel, May 11 2021
  • Mathematica
    Select[Range[3600],Mod[Fibonacci[#]+3,#]==0&] (* Harvey P. Dale, Sep 21 2021 *)

Extensions

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

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

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

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

Programs

  • Maple
    filter:= n -> combinat:-fibonacci(n) - 8 mod n = 0:
select(filter, [$1..10^4]); # Robert Israel, Mar 05 2025
  • Mathematica
    Select[Range[6100], Mod[Fibonacci[#] - 8, #]==0&]  (* Harvey P. Dale, Sep 21 2021 *)

Formula

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

Extensions

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

A078345 Numbers k such that F(k) mod k divides F(F(k) mod k) where F(k) denotes the k-th Fibonacci number.

Original entry on oeis.org

1, 2, 5, 8, 10, 11, 12, 13, 19, 20, 21, 22, 24, 25, 26, 29, 31, 32, 36, 37, 38, 41, 44, 48, 49, 50, 55, 58, 59, 60, 61, 62, 65, 71, 72, 73, 79, 80, 82, 84, 89, 95, 96, 97, 101, 104, 108, 109, 118, 120, 122, 125, 131, 132, 139, 140, 142, 144, 145, 149, 151, 155, 156
Offset: 1

Views

Author

Benoit Cloitre, Nov 22 2002

Keywords

Examples

			F(44) = 701408733; 701408733 mod 44 = 25, F(25)=75025 and 25 divides 75025, hence 44 is in the sequence.

Links

Crossrefs

Cf. A000045, A002708, A023172.

Programs

  • Maple
    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:
filter:= proc(n) local s;
  s:= fmod(n,n);
  fmod(s,s) = 0
end proc:
select(filter, [$1..200]); # Robert Israel, May 10 2016
  • Mathematica
    Unprotect[Divisible];
Divisible[0, 0] = True;
okQ[n_] := Module[{F = Fibonacci, m}, m = Mod[F[n], n];  Divisible[F[m], m]];
Select[Range[75000], okQ] (* Jean-François Alcover, Jul 09 2024 *)

Formula

Conjecture: a(n) is asymptotic to c*n*log(n) with c>0.7

A111419 a(n) is the smallest positive integer for which Fibonacci(n + a(n)) == Fibonacci(n) (mod n).

Original entry on oeis.org

1, 2, 2, 6, 5, 15, 2, 9, 6, 10, 1, 12, 2, 9, 10, 24, 2, 24, 1, 5, 6, 7, 2, 12, 25, 15, 18, 48, 1, 15, 1, 11, 14, 19, 10, 12, 2, 15, 34, 60, 1, 15, 2, 30, 30, 25, 2, 12, 14, 50, 42, 78, 2, 24, 10, 24, 30, 13, 1, 60, 1, 27, 18, 96, 10, 120, 2, 36, 6, 25, 1, 12, 2, 39, 50, 18, 6, 39, 1, 35
Offset: 1

Views

Author

Stefan Steinerberger, Nov 13 2005

Keywords

Comments

When a(n)=2, n is often prime. The exceptions (323, 377, 2834, ...) are in A069107.

Examples

			a(3) = 2 because Fibonacci(3+2) - Fibonacci(3) = 5 - 2 == 0 (mod 3) and 2 is the smallest integer for which this is true.

Links

Crossrefs

Cf. A002708.

Programs

  • Mathematica
    Array[Block[{k = 1}, While[Mod[Fibonacci[# + k], #] != Mod[Fibonacci@ #, #], k++]; k] &, 80] (* Michael De Vlieger, Dec 17 2017 *)
  • MuPAD
    for n from 1 to 100 do an := 0; repeat an := an+1; until (numlib::fibonacci(n+an)-numlib::fibonacci(n)) mod n = 0 end_repeat; print(an); end_for;
  • PARI
    a(n) = {my(k = 1); while(Mod(fibonacci(n + k), n) != Mod(fibonacci(n), n), k++); k;} \\ Michel Marcus, Dec 18 2017

A354443 a(n) = Fibonacci(n^n) mod n.

Original entry on oeis.org

0, 1, 2, 3, 0, 0, 6, 3, 7, 5, 1, 0, 12, 7, 10, 11, 16, 0, 1, 15, 5, 3, 22, 0, 0, 3, 20, 7, 1, 0, 1, 27, 13, 1, 5, 0, 36, 3, 25, 35, 1, 0, 42, 19, 20, 21, 46, 0, 36, 25, 17, 3, 52, 0, 5, 35, 34, 1, 1, 0, 1, 3, 43, 59, 15, 30, 66, 35, 44, 35, 1, 0, 72, 3, 50, 3, 2, 60, 1, 75, 7
Offset: 1

Views

Author

Chittaranjan Pardeshi, May 29 2022

Keywords

Links

Crossrefs

Cf. A000045, A002708, A001175.

Programs

  • PARI
    fibmod(n,m) = lift(((Mod([1,1;1,0], m))^n)[1,2]); \\ as from A001175
a(n) = fibmod(n^n,n);
  • Python
    # uses A001175 python function from A001175
from sympy import fibonacci
def A354443(n): return fibonacci(pow(n,n,A001175(n))) % n # Chai Wah Wu, May 31 2022

A356296 a(n) = Fibonacci(n)^2 mod n.

Original entry on oeis.org

0, 1, 1, 1, 0, 4, 1, 1, 4, 5, 1, 0, 1, 1, 10, 9, 1, 10, 1, 5, 4, 1, 1, 0, 0, 1, 22, 9, 1, 10, 1, 25, 4, 1, 25, 0, 1, 1, 4, 25, 1, 22, 1, 9, 40, 1, 1, 0, 22, 25, 4, 9, 1, 10, 25, 49, 4, 1, 1, 0, 1, 1, 22, 25, 25, 64, 1, 9, 4, 15, 1, 0, 1, 1, 25, 9, 4, 64, 1, 25, 49, 1, 1, 72, 25, 1
Offset: 1

Views

Author

R. J. Mathar, Aug 03 2022

Keywords

Links

Crossrefs

Cf. A000045, A002708, A023172 (location of zeros), A337231, A337232.

Programs

  • Maple
    A356296 := proc(n)
    modp(combinat[fibonacci](n)^2,n) ;
end proc:
seq(A356296(n),n=1..120) ;
  • Mathematica
    Array[PowerMod[Fibonacci[#], 2, #] &, 86] (* Michael De Vlieger, Aug 03 2022 *)
  • PARI
    a(n) = lift(Mod(fibonacci(n), n)^2); \\ Michel Marcus, Aug 03 2022
  • Python
    from sympy import fibonacci
def a(n): return pow(fibonacci(n), 2, n)
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, Aug 04 2022

Formula

a(n) = A000045(n)^2 mod n.

A028253 n mod Fibonacci(n).

Original entry on oeis.org

0, 0, 1, 1, 0, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A002708.

Programs

  • Maple
    with(combinat): [ seq( n mod fibonacci(n), n=1..80) ];
  • Mathematica
    Table[Mod[n,Fibonacci[n]],{n,60}] (* or *) LinearRecurrence[{2,-1},{0,0,1,1,0,6,7},60] (* Harvey P. Dale, Jul 06 2019 *)
  • PARI
    a(n)=if(n>5,n,[0, 0, 1, 1, 0][n]) \\ Charles R Greathouse IV, Aug 22 2011

Formula

a(n) = n for n>5. G.f.: -x^3*(5*x^4-7*x^3+x^2+x-1) / (x-1)^2. - Colin Barker, Aug 05 2013

A224824 Smallest m such that Fibonacci(m) >= m^n.

Original entry on oeis.org

5, 12, 21, 30, 41, 51, 62, 73, 85, 97, 109, 122, 134, 147, 160, 174, 187, 200, 214, 228, 242, 256, 270, 284, 298, 312, 327, 342, 356, 371, 386, 401, 416, 431, 446, 461, 476, 491, 507, 522, 538, 553, 569, 585, 600, 616, 632, 648, 664, 680
Offset: 1

Views

Author

Michel Marcus, Jul 21 2013

Keywords

Crossrefs

Cf. A002708, A132634, A132636.
Cf. A065220, A014283.

Programs

  • Mathematica
    Flatten[{5,Table[Ceiling[(n*LambertW[-1,-Log[GoldenRatio]/(n*5^(1/(2*n)))])/-Log[GoldenRatio]],{n,2,50}]}] (* Vaclav Kotesovec, Jul 24 2013 *)
  • PARI
    a(n) = {my(ok = 0, m = 2); until (ok, if (fibonacci(m) >= m^n, ok = 1, m++); ); return (m); } \\ Michel Marcus, Jul 21 2013; corrected Jun 13 2022

A253109 a(n) = n ^ (Fibonacci(n) mod n).

Original entry on oeis.org

1, 2, 9, 64, 1, 36, 117649, 32768, 4782969, 100000, 11, 1, 23298085122481, 793714773254144, 576650390625, 17592186044416, 48661191875666868481, 3570467226624, 19, 3200000, 4084101, 22, 907846434775996175406740561329, 1, 1, 236773830007967588876795164938469376
Offset: 1

Views

Author

Carlos Eduardo Olivieri, Dec 26 2014

Keywords

Comments

a(n) = 1 for n in A023172. a(n) = n for n in A023173. - Robert Israel, Dec 28 2014

Examples

			a(3) = 9, since 3^(Fibonacci(3) mod 3) = 3^(2 mod 3) = 3^2 = 9.
a(7) = 117649, since 7^(Fibonacci(7) mod 7) = 7^(13 mod 7) = 7^6 = 117649.

Crossrefs

Cf. A000045, A002708, A023172, A023173, A168041.

Programs

  • Magma
    [n^(Fibonacci(n) mod n): n in [1..30]]; // Vincenzo Librandi, Feb 03 2015
  • Maple
    seq(n ^ (combinat:-fibonacci(n) mod n), n=1 .. 50); # Robert Israel, Dec 28 2014
  • Mathematica
    Table[n ^ Mod[Fibonacci[n], n], {n, 29}]
  • PARI
    a(n)=n^(fibonacci(n)%n) \\ Charles R Greathouse IV, Feb 02 2015
Previous Showing 11-20 of 20 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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