OBM 2016 Problem 3

Let it $k$ be a fixed positive integer. Alberto and Beralto play the following game: Given an initial number $N_0$ and starting with Alberto, they alternately do the following operation: change the number $n$ for a number $m$ such that $m < n$ and $m$ and $n$ differ, in its base-2 representation, in exactly $l$ consecutive digits for some $l$ such that $1 \leq l \leq k$. If someone can't play, he loses.

We say a non-negative integer $t$ is a winner number when the gamer who receives the number $t$ has a winning strategy, that is, he can choose the next numbers in order to guarrantee his own victory, regardless the options of the other player. Else, we call it loser.

Prove that, for every positive integer $N$, the total of non-negative loser integers lesser than $2^N$ is $2^{N-\lfloor \frac{log(min\{N,k\})}{log 2} \rfloor}$