|
|
A104714
|
|
Greatest common divisor of a Fibonacci number and its index.
|
|
11
|
|
|
0, 1, 1, 1, 1, 5, 2, 1, 1, 1, 5, 1, 12, 1, 1, 5, 1, 1, 2, 1, 5, 1, 1, 1, 24, 25, 1, 1, 1, 1, 10, 1, 1, 1, 1, 5, 36, 1, 1, 1, 5, 1, 2, 1, 1, 5, 1, 1, 48, 1, 25, 1, 1, 1, 2, 5, 7, 1, 1, 1, 60, 1, 1, 1, 1, 5, 2, 1, 1, 1, 5, 1, 72, 1, 1, 25, 1, 1, 2, 1, 5, 1, 1, 1, 12, 5, 1, 1, 1, 1, 10, 13, 1, 1, 1, 5, 96, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,6
|
|
COMMENTS
|
Considering this sequence is a natural sequel to the investigation of the problem when F_n is divisible by n (the numbers occurring in A023172). This sequence has several nice properties. (1) n | m implies a(n) | a(m) for arbitrary naturals n and m. This property is a direct consequence of the analogous well-known property of Fibonacci numbers. (2) gcd (a(n), a(m)) = a(gcd(n, m)) for arbitrary naturals n and m. Also this property follows directly from the analogous (perhaps not so well-known) property of Fibonacci numbers. (3) a(n) * a(m) | a(n * m) for arbitrary naturals n and m. This property is remarkable especially in the light that the analogous proposition for Fibonacci numbers fails if n and m are not relatively prime (e.g. F_3 * F_3 does not divide F_9). (4) The set of numbers satisfying a(n) = n is closed w.r.t. multiplication. This follows easily from (3).
|
|
LINKS
|
|
|
FORMULA
|
a(n) = gcd(F(n), n).
|
|
EXAMPLE
|
The natural numbers: 0 1 2 3 4 5 6 7 8 9 10 11 12 ...
The Fibonacci numbers: 0 1 1 2 3 5 8 13 21 34 55 89 144 ...
The corresponding GCDs: 0 1 1 1 1 5 2 1 1 1 5 1 12 ...
|
|
MAPLE
|
b:= proc(n) option remember; local r, M, p; r, M, p:=
<<1|0>, <0|1>>, <<0|1>, <1|1>>, n;
do if irem(p, 2, 'p')=1 then r:= r.M mod n fi;
if p=0 then break fi; M:= M.M mod n
od; r[1, 2]
end:
a:= n-> igcd(n, b(n)):
|
|
MATHEMATICA
|
|
|
PROG
|
(Haskell) let fibs@(_ : fs) = 0 : 1 : zipWith (+) fibs fs in 0 : zipWith gcd [1 ..] fs
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
Harmel Nestra (harmel.nestra(AT)ut.ee), Apr 23 2005
|
|
STATUS
|
approved
|
|
|
|