I'm currently working on a problem for which I'd like to include a mathematical definition. I'm an amateur, so please correct me if anything is already wrong.
The set of possible solutions for $x, b \in \mathbb{N}$:
$$M_b = \{x \in \mathbb{N} : l_b(x)=b-1, n \mid \lfloor\frac{x}{b^{b-n-1}}\rfloor\ \forall n \in \mathbb{N} \cap \left[1,b-1\right]\}$$
with $l_b(x)$ being a function that returns the number of digits of a number when written in base $b$.
An additional condition that has to be true for every solution is that no digits repeat. So $1234$ is a possible solution based on that condition alone, while $1223$ is not.
The rest of my formula seems to be working for any $\mathbb{N}_b$ to me, but of course this "digits can't repeat" condition also has to be true for all $b$s (e.g. $\text{AB3F}$ with $b=16$ passes, while $\text{ABB3}$ does not).
Is there any formal way to define such a condition mathematically? Thanks in advance!