* Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor
$$
* Let $T(n,k)$ be a $(k+1)$-th bit from the right side in the binary expansion of $n$. Here
$$
T(n, k) = \left\lfloor\frac{n}{2}\right\rfloor \operatorname{mod} 2
$$
* Let $a(n)$ be a sequence of positive integers such that we start with $A:=n$ and then for $0\leqslant i\leqslant \ell(n)$ apply $A:=A + 2^iT(A,\ell(n)-i)$.

* Let
$$
b(n) = \frac{a(2n)-1}{2} - n
$$

I conjecture that $b(n)$ is a permutation of natural numbers such that first such that first $2^m-1$ terms is a permutation of the first $2^m-1$ natural numbers for $m\in\mathbb{N}$.

Is there a way to prove it?