A074724 Highest power of 3 dividing F(4n) where F(k) is the k-th Fibonacci number.

3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 81, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 81, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 243, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3

Offset

1,1

Comments

If m == 1, 2 or 3 (mod 4) then F(m) is not divisible by 3.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

If k == 1 or 2 (mod 3) then a(3^m*k) = 3^(m+1) for m>=0.

a(n) = A038500(A033888(n)). - Amiram Eldar, May 13 2022

a(n) = 3^A051064(n) (conjectured). - Michel Marcus, May 17 2022

Conjecture: a(n) = (sigma(3*n) - sigma(n))/(sigma(3*n) - 3*sigma(n)), where sigma(n) = A000203(n). Equivalently, a(n) = A088838(n) - A074724(n). - Peter Bala, Jun 10 2022

Mathematica

Table[3^IntegerExponent[Fibonacci[4n], 3], {n, 100}] (* Harvey P. Dale, Jun 03 2012 *)

Prog

(PARI) a(n) = 3^valuation(fibonacci(4*n), 3); \\ Michel Marcus, May 13 2022

Crossrefs

Cf. A000045, A000203, A038500, A033888, A051064, A074724, A088838.

Sequence in context: A353616 A225074 A127795 * A109932 A071909 A134662

Adjacent sequences: A074721 A074722 A074723 * A074725 A074726 A074727

Keyword

nonn

Author

Benoit Cloitre, Sep 04 2002

Status

approved