A235702 -id:A235702 - cp's OEIS frontend

A001175 Pisano periods (or Pisano numbers): period of Fibonacci numbers mod n.

1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60, 16, 30, 48, 24, 100, 84, 72, 48, 14, 120, 30, 48, 40, 36, 80, 24, 76, 18, 56, 60, 40, 48, 88, 30, 120, 48, 32, 24, 112, 300, 72, 84, 108, 72, 20, 48, 72, 42, 58, 120, 60, 30, 48, 96, 140, 120, 136

Offset: 1

N. J. A. Sloane

These numbers might also have been called Fibonacci periods.
Also, number of perfect multi-Skolem type sequences of order n.
Index the Fibonacci numbers so that 3 is the fourth number. If the modulo base is a Fibonacci number (>=3) with an even index, the period is twice the index. If the base is a Fibonacci number (>=5) with an odd index, the period is 4 times the index. - Kerry Mitchell, Dec 11 2005
Each row of the image represents a different modulo base n, from 1 at the bottom to 24 at the top.  The columns represent the Fibonacci numbers mod n, from 0 mod n at the left to 59 mod n on the right.  In each cell, the brightness indicates the value of the residual, from dark for 0 to near-white for n-1.  Blue squares on the left represent the first period; the number of blue squares is the Pisano number. - Kerry Mitchell, Feb 02 2013
a(n) = least positive integer k such that F(k) == 0 (mod n) and F(k+1) == 1 (mod n), where F = A000045 is the Fibonacci sequence. a(n) exists for all n by Dirichlet's box principle and the fact that the positive integers are well-ordered. Cf. Saha and Karthik (2011). - L. Edson Jeffery, Feb 12 2014

			For n=4, take the Fibonacci sequence (A000045), 1, 1, 2, 3, 5, 8, 13, 21, ... (mod 4), which gives 1, 1, 2, 3, 1, 0, 1, 1, .... This repeats a pattern of length 6, so a(4) = 6. - _Michael B. Porter_, Jul 19 2016

B. H. Hannon and W. L. Morris, Tables of Arithmetical Functions Related to the Fibonacci Numbers. Report ORNL-4261, Oak Ridge National Laboratory, Oak Ridge, Tennessee, Jun 1968.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 162.
J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, pp. 304 - 309.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989. See p. 89. - N. J. A. Sloane, Feb 20 2013

T. D. Noe, Table of n, a(n) for n = 1..10000
B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and Journal of Integer Sequences 17 (2014), #14.8.5.
Michael Baake, Natascha Neumarker, and John A. G. Roberts, Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices, arXiv:1205.1003 [math.DS], 2012.
Brennan Benfield and Michelle Manes, The Fibonacci Sequence is Normal Base 10, arXiv:2202.08986 [math.NT], 2022.
K. S. Brown, Periods of Fibonacci Sequences mod m.
K. S. Brown, Periods of Fibonacci Sequences mod m. [Cached copy in pdf format]
Francis N. Castro and Luis A. Medina, Modular periodicity of exponential sums of symmetric Boolean functions and some of its consequences, arXiv:1603.00534 [math.NT], 2016.
D. A. Coleman, C. J. Dugan, R. A. McEwen, C. A. Reiter, and T. T. Tang, Periods of (q,r)-Fibonacci sequences and Elliptic Curves, Fibonacci Quart. 44 (1) (2006), 59-70.
Joseph Louis de Lagrange, Additions aux éléments d'algèbre d'Euler. Analyse indéterminée. (1774), pp. 143ff.
H. T. Engstrom, On sequences defined by linear recurrence relations, Trans. Am. Math. Soc. 33 (1) (1931), 210-218.
J. D. Fulton and W. L. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arithmetica 16 (1969), 105-110.
José María Grau and Antonio M. Oller-Marcén, On the last digit and the last non-zero digit of n^n in base b, arXiv preprint arXiv:1203.4066 [math.NT], 2012.
James Grime and Brady Haran, Fibonacci Mystery, Numberphile video, 2013.
B. H. Hannon and W. L. Morris, Tables of Arithmetical Functions Related to the Fibonacci Numbers. [Annotated and scanned copy]
Naoki Sato, Cyrus Hsia, Richard Hoshino, Wai Ling Yee, and Adrian Chan, editors, Fibonacci residues, Crux Mathematicorum 23:4 (1997), pp. 224-226.
Dan Ismailescu and Peter C. Shim, On numbers that cannot be expressed as a plus-minus weighted sum of a Fibonacci number and a prime, INTEGERS 14 (2014), #A65.
Kerry Mitchell, Illustration of Pisano Numbers as Periods of Fibonacci Numbers Mod n
G. Nordh, Perfect Skolem sequences, arXiv:math/0506155 [math.CO], 2005.
Noel Patson, Pisano period and permutations of n X n matrices, Australian Math. Soc. Gazette, 2007.
Noel Patson, Square Matrix Permutations.
M. Renault, Periods of Fibonacci Sequence Modulo m.
Arpan Saha and C. S. Karthik, A few equivalences of [the] Wall-Sun-Sun prime conjecture, p. 2, arXiv:1102.1636 [math.NT], 2011.
Minjia Shi, Zhongyi Zhang, and Patrick Solé, Pisano period codes, arXiv:1709.04582 [cs.IT], 2017.
D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.
Eric Weisstein's World of Mathematics, Pisano Number.
Wikipedia, Pisano period.
J. W. W. [J. W. Wrench], Review of B. H. Hannon and W. L. Morris tables, Math. Comp., 23 (1969), 459-460.

Cf. A060305 (Fibonacci period mod prime(n)), A003893.

Cf. A001178 (Fibonacci frequency), A001179 (Leonardo logarithm), A235702 (fixed points), A066853 (number of elements in the set of residua), A222413, A296240 (Pisano quotients for primes).

a001175 1 = 1
a001175 n = f 1 ps 0 where
   f 0 (1 : xs) pi = pi
   f _ (x : xs) pi = f x xs (pi + 1)
   ps = 1 : 1 : zipWith (\u v -> (u + v) `mod` n) (tail ps) ps
-- Reinhard Zumkeller, Jan 15 2014

a:= proc(n) local f, k, l; l:= ifactors(n)[2];
      if nops(l)<>1 then ilcm(seq(a(i[1]^i[2]), i=l))
    else f:= [0, 1];
         for k do f:=[f[2], f[1]+f[2] mod n];
                  if f=[0, 1] then break fi
         od; k
      fi
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 18 2013

Table[a={1, 0}; a0=a; k=0; While[k++; s=Mod[Plus@@a, n]; a=RotateLeft[a]; a[[2]]=s; a!=a0]; k, {n, 2, 100}] (* T. D. Noe, Jul 19 2005 *)
a[1]=1; a[n_]:= For[k = 1, True, k++, If[Mod[Fibonacci[k], n]==0 && Mod[ Fibonacci[k+1], n]==1, Return[k]]]; Table[a[n], {n, 100}] (* Jean-François Alcover, Feb 11 2015 *)
test[{0, 1, }]:= False; test[]:= True;
nest[k_][{a_, b_, c_}]:= {Mod[b, k], Mod[a+b, k], c+1};
A001175[1] := 1;
A001175[k_]:= NestWhile[nest[k], {1, 1, 1}, test][[3]];
Table[A001175[n], {n, 100}] (* Leo C. Stein, Nov 08 2019 *)
Module[{nn=1000,fibs},fibs=Fibonacci[Range[nn]];Table[Length[ FindTransientRepeat[ Mod[fibs,n],2][[2]]],{n,70}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 02 2020 *)

minWSS=2^64; \\ PrimeGrid search
fibmod(n,m)=((Mod([1,1;1,0],m))^n)[1,2]
entryp(p)=my(k=p+[0, -1, 1, 1, -1][p%5+1], f=factor(k)); for(i=1, #f[, 1], for(j=1, f[i, 2], if((Mod([1, 1; 1, 0], p)^(k/f[i, 1]))[1, 2], break); k/=f[i, 1])); k
entry(n)=if(n==1,return(1)); my(f=factor(n), v); v=vector(#f~, i, if(f[i,1]>=minWSS, entryp(f[i,1]^f[i,2]), entryp(f[i,1])*f[i,1]^(f[i,2] - 1))); if(f[1,1]==2&&f[1,2]>1, v[1]=3<Charles R Greathouse IV, Feb 13 2014; updated Dec 14 2016; updated Aug 24 2021; updated Jul 08 2024

from functools import reduce
from sympy import factorint, lcm
def A001175(n):
    if n == 1:
        return 1
    f = factorint(n)
    if len(f) > 1:
        return reduce(lcm, (A001175(a**f[a]) for a in f))
    else:
        k,x = 1, [1,1]
        while x != [0,1]:
            k += 1
            x = [x[1], (x[0]+x[1]) % n]
        return k # Chai Wah Wu, Jul 17 2019

def a(n): return BinaryRecurrenceSequence(1,1).period(n) # Ralf Stephan, Jan 23 2014

Let the prime factorization of n be p1^e1...pk^ek. Then a(n) = lcm(a(p1^e1), ..., a(pk^ek)). - T. D. Noe, May 02 2005
a(n) = n-1 if n is a prime > 5 included in A003147 ( n = 11, 19, 31, 41, 59, 61, 71, 79, 109...). - Benoit Cloitre, Jun 04 2002
K. S. Brown shows that a(n)/n <= 6 for all n and a(n)=6n if and only if n has the form 2*5^k.
a(n) = A001177(n)*A001176(n) for n >= 1. - Henry Bottomley, Dec 19 2001
a(n) <= 2*n+2 if n is a prime > 5, by Wall's Theorems 6 and 7; see A060305, A222413, A296240. - Jonathan Sondow, Dec 10 2017
a(2^k) = 3*2^(k-1) for k > 0. In general, if a(p) != a(p^2) for p prime, then a(p^k) = p^(k-1)*a(p) [Wall, 1960]. - Chai Wah Wu, Feb 25 2022

A001178 Fibonacci frequency of n.

Original entry on oeis.org

0, 4, 3, 2, 3, 1, 2, 2, 1, 2, 3, 1, 3, 2, 3, 1, 2, 1, 2, 2, 2, 2, 2, 0, 3, 3, 2, 2, 3, 1, 2, 2, 3, 2, 2, 1, 3, 2, 3, 2, 3, 2, 3, 2, 1, 2, 3, 1, 3, 2, 2, 3, 3, 2, 3, 2, 2, 3, 4, 1, 2, 2, 2, 3, 3, 1, 3, 2, 2

Offset: 1

N. J. A. Sloane

nonn

a(A235702(n)) = 0. - Reinhard Zumkeller, Jan 15 2014

a(n) is the least nonnegative integer k such that the function iterates f: {1, 2, ...} -> {1, 2, ...}, n -> f(n) = A001175(n), satisfy f^[k+1](n) = f^[k](n), where f^[0] is the identity map f^[0](n) = n and f^[k+1] = f o f^[k]. See the Fulton and Morris link, where the function f is called pi and a(n)= omega(n) for n >= 2, and omega(24) should be 0. (see the Zumkeller remark on the Hannon and Morris reference) - Wolfdieter Lang, Jan 18 2015

B. H. Hannon and W. L. Morris, Tables of Arithmetical Functions Related to the Fibonacci Numbers. Report ORNL-4261, Oak Ridge National Laboratory, Oak Ridge, Tennessee, Jun 1968. [There is a typo in the value of a(24) given in the table on the last page.]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. Fulton and W. L. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arithmetica, 16 (1969), 105-110.
B. H. Hannon and W. L. Morris, Tables of Arithmetical Functions Related to the Fibonacci Numbers [Annotated and scanned copy]
Wikipedia, Pisano period

Cf. A001175, A235702.

a001178 = f 0 where
   f j x = if x == y then j else f (j + 1) y  where y = a001175 x
-- Reinhard Zumkeller, Jan 15 2014

pi[1] = 1;
pi[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0 && Mod[ Fibonacci[k+1], n] == 1, Return[k]]];
a[n_] := Length[FixedPointList[pi, n]] - 2;
a /@ Range[100] (* Jean-François Alcover, Oct 28 2019 *)

from itertools import count
def A001178(n): # uses function from A001175
    m = n
    for c in count(0):
        k = A001175(m)
        if k == m:
            return c
        m = k # Chai Wah Wu, Feb 28 2022

See a comment above and the program.

a(24) corrected by Reinhard Zumkeller, Jan 15 2014

A379646 Irregular triangle T(n,k) where row n contains the trajectory of recursive mappings of A001175(x) starting with x = n and ending at fixed point A235249(n).

Original entry on oeis.org

1, 2, 3, 8, 12, 24, 3, 8, 12, 24, 4, 6, 24, 5, 20, 60, 120, 6, 24, 7, 16, 24, 8, 12, 24, 9, 24, 10, 60, 120, 11, 10, 60, 120, 12, 24, 13, 28, 48, 24, 14, 48, 24, 15, 40, 60, 120, 16, 24, 17, 36, 24, 18, 24, 19, 18, 24, 20, 60, 120, 21, 16, 24, 22, 30, 120, 23, 48, 24

Offset: 1

Michael De Vlieger, Dec 30 2024

Row n contains recursive mappings of A001175(x) starting with x = n.

			Table begins:
   1;
   2,  3,   8,  12, 24;
   3,  8,  12,  24;
   4,  6,  24;
   5, 20,  60, 120;
   6, 24;
   7, 16,  24;
   8, 12,  24;
   9, 24;
  10, 60, 120;
  11, 10,  60, 120;
  12, 24;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..10141 (rows n = 1..2500, flattened)
Brennan Benfield and Oliver Lippard, Fixed points of K-Fibonacci sequences, arXiv:2404.08194 [math.NT], 2024.
J. D. Fulton and W. L. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arithmetica, 16 (1969), 105-110.
Wikipedia, Pisano period

Cf. A001175, A001179, A235249, A235702.

q[{0, 1, }] := False; q[] := True;
f[k_][{a_, b_, c_}] := {Mod[b, k], Mod[a + b, k], c + 1};
s[1] := 1; s[k_] := s[k] = Which[
  PrimeQ[k] && k > 5, If[
    AnyTrue[PrimitiveRootList[k], Mod[#^2, k] == Mod[# + 1, k] &],
    k - 1,
    NestWhile[f[k], {1, 1, 1}, q][[-1]] ],
  PrimePowerQ[k], NestWhile[f[k], {1, 1, 1}, q][[-1]], True,
    LCM @@ Map[s[#] &, Power @@@ FactorInteger[k] ] ];
Table[Most@ FixedPointList[s[#] &, n], {n, 24}]

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A001175 Pisano periods (or Pisano numbers): period of Fibonacci numbers mod n.

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A001178 Fibonacci frequency of n.

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A379646 Irregular triangle T(n,k) where row n contains the trajectory of recursive mappings of A001175(x) starting with x = n and ending at fixed point A235249(n).

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A343184 Positions of records in A001178.

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