|
| |
|
|
A001176
|
|
Number of zeros in fundamental period of Fibonacci numbers mod n.
(Formerly M0165 N0064)
|
|
12
| |
|
|
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; 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 (lkmitch(AT)gmail.com), Dec 11 2005
|
|
|
REFERENCES
| 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. Report ORNL-4261, Oak Ridge National Laboratory, Oak Ridge, Tennessee, Jun 1968.
Review of B. H. Hannon and W. L. Morris tables, Math. Comp., 23 (1969), 459-460.
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
| T. D. Noe, Table of n, a(n) for n=1..1000
M. Renault, Fibonacci sequence modulo m
|
|
|
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. [From 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.
|
|
|
CROSSREFS
| Cf. A001175, A001177, A053027, A053028, A053029, A053030, A053031, A053032.
Sequence in context: A109090 A080100 A161822 * A136693 A086685 A094571
Adjacent sequences: A001173 A001174 A001175 * A001177 A001178 A001179
|
|
|
KEYWORD
| nonn,easy
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| Better description and more terms from Henry Bottomley (se16(AT)btinternet.com), Feb 01 2000. Examples from David W. Wilson, Jan 05 2005.
|
| |
|
|
