MathDB

Arnaldo and Bernaldo play a game where they alternate saying natural numbers, and the winner is the one who says

0

. In each turn except the first the possible moves are determined from the previous number

n

in the following way: write

n =\sum_{m\in O_n}2^m;

the valid numbers are the elements

m

O_n

. That way, for example, after Arnaldo says

42= 2^5 + 2^3 + 2^1

, Bernaldo must respond with

5

3

1

.
We define the sets

A,B\subset \mathbb{N}

in the following way. We have

n\in A

iff Arnaldo, saying

n

in his first turn, has a winning strategy; analogously, we have

n\in B

iff Bernaldo has a winning strategy if Arnaldo says

n

during his first turn. This way,

A =\{0, 2, 8, 10,\cdots\}, B = \{1, 3, 4, 5, 6, 7, 9,\cdots\}

Define

f:\mathbb{N}\to \mathbb{N}

f(n)=|A\cap \{0,1,\cdots,n-1\}|

. For example,

f(8) = 2

and

f(11)=4

. Find

\lim_{n\to\infty}\frac{f(n)\log(n)^{2005}}{n}

5