Given a number $x \in \mathbb{N}$ , I want to write down following algorithm in a notation which can be written without the need for providing an example.
Step (1): Find all unique prime factors $p_1, p_2, ..., p_n$ of $x$ ( $p_1 \neq p_2 \neq ... \neq p_n$ , $p_n \in \mathbb{P}$ , $n \in \mathbb{N}$ ):
$x = p_1^{u_1} \cdot p_2^{u_2} \cdot ... \cdot p_n^{u_n}$
Step (2): ???
Example for $x:=10$
$x := 10 = 2^1 \cdot 5^1 \Rightarrow p_1 = 2$ , $p_2 = 5$
$D_1(10) = \{ m | m \equiv 0\ (mod\ 2) \wedge m \equiv 0\ (mod\ 5) , m \in \mathbb{N}\}$
$D_2(10) = \{ m | m \equiv 0\ (mod\ 2) \wedge m \equiv 1\ (mod\ 5) , m \in \mathbb{N}\}$
$D_3(10) = \{ m | m \equiv 1\ (mod\ 2) \wedge m \equiv 0\ (mod\ 5) , m \in \mathbb{N}\}$
$D_4(10) = \{ m | m \equiv 1\ (mod\ 2) \wedge m \equiv 1\ (mod\ 5) , m \in \mathbb{N}\}$
Example for $x:=30$
$x := 30 = 2^1 \cdot 3^1 \cdot 5^1 \Rightarrow p_1 = 2$ , $p_2 = 3$ , $p_3 = 5$
$D_1(30) = \{ m | m \equiv 0\ (mod\ 2) \wedge m \equiv 0\ (mod\ 3) \wedge m \equiv 0\ (mod\ 5) , m \in \mathbb{N}\}$
$D_2(30) = \{ m | m \equiv 0\ (mod\ 2) \wedge m \equiv 0\ (mod\ 3) \wedge m \equiv 1\ (mod\ 5) , m \in \mathbb{N}\}$
$D_3(30) = \{ m | m \equiv 0\ (mod\ 2) \wedge m \equiv 1\ (mod\ 3) \wedge m \equiv 0\ (mod\ 5) , m \in \mathbb{N}\}$
$D_4(30) = \{ m | m \equiv 0\ (mod\ 2) \wedge m \equiv 1\ (mod\ 3) \wedge m \equiv 1\ (mod\ 5) , m \in \mathbb{N}\}$
$D_5(30) = \{ m | m \equiv 1\ (mod\ 2) \wedge m \equiv 0\ (mod\ 3) \wedge m \equiv 0\ (mod\ 5) , m \in \mathbb{N}\}$
$D_6(30) = \{ m | m \equiv 1\ (mod\ 2) \wedge m \equiv 0\ (mod\ 3) \wedge m \equiv 1\ (mod\ 5) , m \in \mathbb{N}\}$
$D_7(30) = \{ m | m \equiv 1\ (mod\ 2) \wedge m \equiv 1\ (mod\ 3) \wedge m \equiv 0\ (mod\ 5) , m \in \mathbb{N}\}$
$D_8(30) = \{ m | m \equiv 1\ (mod\ 2) \wedge m \equiv 1\ (mod\ 3) \wedge m \equiv 1\ (mod\ 5) , m \in \mathbb{N}\}$
For step (2) I have problems describing this binary permutation in natural language, and in a mathematical notation, without providing an example. I am searching for a mathematical optimal notation and easy understandable description. I would be glad if you could help me with this.