A083523 Smallest Fibonacci number divisible by 2^n.

1, 2, 8, 8, 144, 46368, 4807526976, 51680708854858323072, 5972304273877744135569338397692020533504, 79757008057644623350300078764807923712509139103039448418553259155159833079730688

Offset: 0

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 05 2003

Keywords

• nonn

Comments

The index of the Fibonacci numbers above begin: 1, 3, 6, 6 and then doubles thereafter.

Links

• Amiram Eldar, Table of n, a(n) for n = 0..12
• Ron Knott, The first 300 Fibonacci numbers, completely factorised.
• Tamás Lengyel, The order of the Fibonacci and Lucas numbers, The Fibonacci Quarterly, Vol. 33, No. 3 (1995), pp. 234-239.

Crossrefs

Cf. A000045, A001622, A007283, A079613, A337923.

Programs

Mathematica

Do[k = 1; While[ !IntegerQ[ Fibonacci[k]/2^n], k++ ]; Print[ Fibonacci[k]], {n, 0, 10}]
With[{fibs=Fibonacci[Range[1000]]},Table[SelectFirst[fibs, Divisible[#,2^n]&],{n,0,10}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 02 2021 *)
Join[{1, 2, 8}, Table[Fibonacci[3*2^(n - 2)], {n, 3, 9}]] (* Amiram Eldar, Jan 29 2022 *)

Formula

From Amiram Eldar, Jan 29 2022: (Start)

a(n) = Fibonacci(3*2^(n-2)) = A000045(A007283(n-2)) = A079613(n-2), for n > 2.

Sum_{n>=0} 1/a(n) = 19/8 - 1/phi, where phi is the golden ratio (A001622). (End)

Extensions

Edited and extended by Robert G. Wilson v, May 06 2003