Coefficients of some polynomials factorization

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EDITED

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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?

I would encourage you to provide some explicit examples, in part because your writeup is too terse to express you ideas. Some issues i have are: 1/ You used \cdot in the polynomial factorization of $P_{n_b}$. So, are you referring to a specific polynomial factorization, which could be $ 1 \times P_{n_b}$, the fully factorized form, or any possible factorization? 2/ Your post is for general $b$, but the link for "iff no carries" is only for $b = 2$. Did you verify the general $b$ case? 3/ Please clarify what you mean by $ -10 \leq b $. Are you using negative bases? (You might be) non-integer? — Calvin Lin, Feb 28 '24 at 15:26

Coefficients of some polynomials factorization

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