A083523 - OEIS

%I #17 Jan 29 2022 04:56:23

%S 1,2,8,8,144,46368,4807526976,51680708854858323072,

%T 5972304273877744135569338397692020533504,

%U 79757008057644623350300078764807923712509139103039448418553259155159833079730688

%N Smallest Fibonacci number divisible by 2^n.

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

%H Amiram Eldar, <a href="/A083523/b083523.txt">Table of n, a(n) for n = 0..12</a>

%H Ron Knott, <a href="https://r-knott.surrey.ac.uk/Fibonacci/fibtable.html#100">The first 300 Fibonacci numbers, completely factorised</a>.

%H Tamás Lengyel, <a href="https://www.fq.math.ca/Scanned/33-3/lengyel.pdf">The order of the Fibonacci and Lucas numbers</a>, The Fibonacci Quarterly, Vol. 33, No. 3 (1995), pp. 234-239.

%F From _Amiram Eldar_, Jan 29 2022: (Start)

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

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

%t Do[k = 1; While[ !IntegerQ[ Fibonacci[k]/2^n], k++ ]; Print[ Fibonacci[k]], {n, 0, 10}]

%t 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 *)

%t Join[{1, 2, 8}, Table[Fibonacci[3*2^(n - 2)], {n, 3, 9}]] (* _Amiram Eldar_, Jan 29 2022 *)

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

%K nonn

%O 0,2

%A _Amarnath Murthy_ and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 05 2003

%E Edited and extended by _Robert G. Wilson v_, May 06 2003