Characterization of the a-sequence by complexity and the set A

Let $\|n\|$ denote the complexity of a natural number, and define

$$A=\{n\in\mathbb N:\|3^j n\|=3j+\|n\|\text{ for all }j\}.$$

Let $(a_\alpha)_{\alpha<\xi}$ be the transfinite sequence associated with numbers of complexity divisible by $3$. a-sequence characterization conjecture. The terms of $(a_\alpha)$ are exactly the elements of

$$\left\{\frac{n}{3^{\|n\|/3}}:\|n\|\equiv0\pmod 3,\ n\in A\right\},$$

ordered decreasingly.

This gives an intrinsic description of the first sequence in terms of the set $A$ and the complexity of its elements; the paper offers the claim as an unproved conjecture.