A235127 Greatest k such that 4^k divides n.
0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1
Offset: 1
Examples
Since 4^2 divides 32 and 4^3 does not, we have a(32) = 2. Likewise, since no positive power of 4 divides 9, a(9) = 0.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..65537
Programs
-
Mathematica
IntegerExponent[Range@ 105, 4] (* Michael De Vlieger, Nov 18 2017 *)
-
PARI
A235127(n) = valuation(n,4); \\ Antti Karttunen, Nov 18 2017
-
Python
def A235127(n): return (~n&n-1).bit_length()>>1 # Chai Wah Wu, Jul 08 2022
-
Sage
n=100 #change n for more terms [valuation(i,4) for i in [1..n]]
Formula
a(n) = valuation(n,4).
G.f.: Sum_{k>=1} x^(4^k)/(1 - x^(4^k)). - Ilya Gutkovskiy, Jan 28 2017
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/3. - Amiram Eldar, Jan 17 2022
Extensions
More terms from Antti Karttunen, Nov 18 2017