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

Author

Benoit Cloitre, Sep 04 2002

Keywords

• nonn

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

Crossrefs

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

Programs

Mathematica

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

PARI

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

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