A001178 Fibonacci frequency of n.

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

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

References

• 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).

Links

• 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

Crossrefs

Cf. A001175, A235702.

Programs

Haskell

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

Mathematica

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 *)

Python

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

Formula

See a comment above and the program.

Extensions

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