login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001176 Number of zeros in fundamental period of Fibonacci numbers mod n.
(Formerly M0165 N0064)
21
1, 1, 2, 1, 4, 2, 2, 2, 2, 4, 1, 2, 4, 2, 2, 2, 4, 2, 1, 2, 2, 1, 2, 2, 4, 4, 2, 2, 1, 2, 1, 2, 2, 4, 2, 2, 4, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 1, 1, 2, 4, 1, 2, 2, 4, 2, 2, 2, 2, 2, 1, 2, 4, 4, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 1, 2, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

If the Fibonacci numbers are indexed so that 3 is the fourth number, then if the modulo base is a Fibonacci number (>= 3) with an even index, the period has 2 zeros. If the base is a Fibonacci number (>= 5) with an odd index, the period has 4 zeros. - Kerry Mitchell, Dec 11 2005

a(n) = A128924(n,1). - Reinhard Zumkeller, Jan 17 2014

For a proof that A001177(n) divides the period length A001175(n) for n >= 1, see, e.g., the Vajda reference, p. 73. This comment refers to the present first formula. - Wolfdieter Lang, Jan 19 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.

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.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

J. 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]

M. Renault, Fibonacci sequence modulo m

Review of B. H. Hannon and W. L. Morris tables, Math. Comp., 23 (1969), 459-460.

FORMULA

a(n) = A001175(n)/A001177(n) for n >= 1.

a(n) = ord(n, fibonacci(A001177(n) + 1)), where ord(n, a) is the multiplicative order of a modulo n. - Mircea Merca, Jan 03 2011

EXAMPLE

{F(n) mod 1} has fundamental period (0) with 1 zero.

{F(n) mod 2} has fundamental period (0,1,1) with 1 zero.

{F(n) mod 3} has fundamental period (0,1,1,2,0,2,2,1) with 2 zeros.

{F(n) mod 4} has fundamental period (0,1,1,2,3,1), with 1 zero.

{F(n) mod 5} has fundamental period (0,1,1,2,3,0,3,3,1,4,0,4,4,3,2,0,2,2,4,1) with 4 zeros.

MATHEMATICA

With[{fibs=Fibonacci[Range[2000]]}, Table[Count[FindTransientRepeat[ Mod[ fibs, n], 3][[2]], 0], {n, 110}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 26 2016 *)

PROG

(Haskell)

a001176 1 = 1

a001176 n = f 1 ps 0 where

   f 0 (1 : xs) z = z

   f _ (x : xs) z = f x xs (z + 0 ^ x)

   ps = 1 : 1 : zipWith (\u v -> (u + v) `mod` n) (tail ps) ps

-- Reinhard Zumkeller, Jan 15 2014

CROSSREFS

Cf. A001175, A001177, A053027, A053028, A053029, A053030, A053031, A053032.

Cf. A235715.

Sequence in context: A220780 A080100 A161822 * A136693 A086685 A094571

Adjacent sequences:  A001173 A001174 A001175 * A001177 A001178 A001179

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Better description and more terms from Henry Bottomley, Feb 01 2000

Examples from David W. Wilson, Jan 05 2005

Replaced the old Renault link with a working one. - Wolfdieter Lang, Jan 17 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 24 00:24 EDT 2017. Contains 283983 sequences.