EDITED
If anyone is interested in this question, now there are OEIS draft and repository with detailed data.
Let $n,b\in\mathbb{N}$ and $P_{n_b}$ is polynomial whose coefficients are digits of $n$ in base $b$.
Let $Q_{1}(x) \cdot Q_{2}(x) \cdots Q_{m}(x)$ - polynomial factorization (over integers) of $P_{n_b}$.
I'm interested in coefficients of such factorization, especially union of all factor's coefficients $\mathcal{U}(n,b)$ of a given polynomial.
Let $n$ successively increase from $1$. From the previous responses I learned that:
$\mathcal{U}(n,b) \subseteq \{0, \dots, b-1\}$
iff there are no carries when multiplying prime factors of $n$ (in base $b$)Any integer sooner or later will appear in some $\mathcal{U}(n,b)$
I've found numerically that there are certain patterns in the order
in which integer $z$ firstly appear in $\mathcal{U}(n,b)$
E.g. for $b=2$:
$1,0,-1,2,-2,3,-3,\{-4,4\},\{-5,5\},\{-6,6\},\{-7,7\},\{-8,8\},\dots$
(starting with $-4$ integers appear in pairs simultaneously)
Part of table for $b=10$:
| z | 1 | 2 | .. | 9 | 10 | -1 | -2 | .. | 10 | -9 |
| n | 1 | 2 | .. | 9 | 0 | 1001 | 1008 | .. | 28119 | 108009 |
| z | -10 | -11 | .. | 18 | -14 | .. |
| n | 621489 | 730158 | .. | 10809909 | 10951149 | .. |
Are there any algebra or number theories laws behind these patterns, or it's just kind of misconception?