|
Search: id:A104714
|
|
|
| A104714 |
|
Greatest common divisor of a Fibonacci number and its index. |
|
+0
3
|
|
| 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; listen)
|
|
|
OFFSET
|
0,6
|
|
|
COMMENT
|
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).
|
|
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 ...
|
|
PROGRAM
|
(Haskell) let fibs@(_ : fs) = 0 : 1 : zipWith (+) fibs fs in 0 : zipWith gcd [1 ..] fs
|
|
CROSSREFS
|
Cf. A023172, A000045, A001177, A001175, A001176. a(n) = gcd(A000045(n), A001477(n)). a(n) = n iff n occurs in A023172 iff n | A000045(n).
Sequence in context: A113103 A033325 A126690 this_sequence A085119 A010128 A029764
Adjacent sequences: A104711 A104712 A104713 this_sequence A104715 A104716 A104717
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
Harmel Nestra (harmel.nestra(AT)ut.ee), Apr 23 2005
|
|
|
Search completed in 0.002 seconds
|
