

A105953


Numbers n such that the period length P(n) of the Fibonacci sequence mod n is a multiple of n.


0



1, 5, 6, 10, 12, 20, 24, 25, 30, 50, 60, 100, 120, 125, 150, 250, 300, 500, 600, 625, 750, 1250, 1500, 2500, 3000, 3125, 3750, 6250, 7500, 12500, 15000, 15625, 18750, 31250, 37500, 62500, 75000, 78125, 93750, 156250, 187500, 312500, 375000, 390625
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Numbers n such that A001175(n) (mod n) == 0.


LINKS

Table of n, a(n) for n=1..44.


FORMULA

Numbers of the form 2^a*5^b*6^c, b={0, 1, 2, ...}, c={0, 1} and a={0, 1, 2} but only equal 1 or 2 if b or c > 0.  Robert G. Wilson v
Equivalently, [1,6,10,12,20,24]*5^m, m>=0.


EXAMPLE

6 is in the list because the first 24 Fibonacci numbers (A000045) are:
0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657;
mod(Fibonacci(k=0..inf.), 6) is:
0,1,1,2,3,5,2,1,3,4,1,5,0,5,5,4,3,1,4,5,3,2,5,1 repeated;
this has period 24 which is a multiple of 6; therefore 6 is a member.


MATHEMATICA

f[n_] := Block[{a = a0 = {1, 0}, k = 0}, While[k++; s = Mod[Plus @@ a, n]; a = RotateLeft[a]; a[[2]] = s; a != a0]; k]; lst = {1}; Do[ If[ Mod[ f[n], n] == 0, AppendTo[lst, n]; Print[n]], {n, 2, 469000}] (* Robert G. Wilson v, May 31 2005 *)


CROSSREFS

Cf. A001175, A000045.
Sequence in context: A099538 A093614 A093509 * A164095 A102506 A062845
Adjacent sequences: A105950 A105951 A105952 * A105954 A105955 A105956


KEYWORD

nonn


AUTHOR

James HighamKessler (James_HighamKessler(AT)Brown.edu), Apr 27 2005


EXTENSIONS

Edited by Robert G. Wilson v, who also found a(30)a(45), May 31 2005


STATUS

approved



