%I #20 Mar 01 2023 02:36:57

%S 0,0,1,0,0,3,0,0,1,0,0,4,0,0,1,0,0,3,0,0,1,0,0,5,0,0,1,0,0,3,0,0,1,0,

%T 0,4,0,0,1,0,0,3,0,0,1,0,0,6,0,0,1,0,0,3,0,0,1,0,0,4,0,0,1,0,0,3,0,0,

%U 1,0,0,5,0,0,1,0,0,3,0,0,1,0,0,4,0,0,1

%N a(n) is the exponent of the highest power of 2 dividing the n-th Fibonacci number.

%H Amiram Eldar, <a href="/A337923/b337923.txt">Table of n, a(n) for n = 1..10000</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 a(n) = A007814(A000045(n)).

%F The following 4 formulas completely specify the sequence (Lengyel, 1995):

%F 1. a(n) = 0 if n == 1 (mod 3) or n == 2 (mod 3).

%F 2. a(n) = 1 if n == 3 (mod 6).

%F 3. a(n) = 3 if n == 6 (mod 12).

%F 4. a(n) = A007814(n) + 2 if n == 0 (mod 12).

%F a(A001651(n)) = 0.

%F a(A016945(n)) = 1.

%F a(A017593(n)) = 3.

%F a(A073762(n)) = 4.

%F The image of this function is A184985, i.e., all the nonnegative integers excluding 2.

%F Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = 5/6.

%F a(3*n) = A090740(n), a(3*n+1) = a(3*n+2) = 0. - _Joerg Arndt_, Mar 01 2023

%e a(1) = 0 since Fibonacci(1) = 1 is odd.

%e a(6) = 3 since Fibonacci(6) = 8 = 2^3.

%e a(12) = 4 since Fibonacci(12) = 144 = 2^4 * 3^2.

%t a[n_] := IntegerExponent[Fibonacci[n], 2]; Array[a, 100]

%o (Python)

%o def A337923(n): return int(not n%3)+(int(not n%6)<<1) if n%12 else 2+(~n&n-1).bit_length() # _Chai Wah Wu_, Jul 10 2022

%Y Cf. A000045, A007814, A248174.

%Y Cf. A001651, A016945, A017593, A073762, A184985.

%Y Cf. A090740 (sequence without zeros).

%K nonn

%O 1,6

%A _Amiram Eldar_, Jan 29 2021