In the study of number theory, odd and even numbers are a fundamental pair of ideas. Natural number sets come in two different varieties. Numerological theory frequently examines the connections between various numbers. There are numerous conjectures that attempt to generalize the law of different sorts of natural numbers discovered in a restricted range to the entire infinite set of natural numbers. This article will examine the famous Collatz conjecture, which states that for each natural number n, if it is even, divide by 2, if it is odd, multiply by 3, add 1, and so on, the result must finally reach 1. It is also referred to as the $3n+1$ conjecture and was put forth in 1937 by Lothar Collatz, also known as the $3n+1$ problem. The mathematician Paul Erdos once said of this conjecture: "Mathematics may not be ready for such problems"${}^{[1,2]}$.

The inconsistencies between the finite and the infinite, as well as the relation between various kinds, present difficulties in the study of number theory problems. We are talking about the connection between two different mathematical ideas: the iterated sequence is a ultimately periodic sequence whether the initial value is odd or even.

The finite and the infinite can be connected by the useful mathematical construct known as a function, and the resulting outcomes will also be finite and infinite. A special function that has particular significance in discrete mathematics is the piecewise function. Compound functions and piecewise functions combined is a highly clever mathematical trick. Particularly in number theory, which is the most fundamental idea and mathematical expression, the sequence of numbers is a close connection between functions and finite and infinite. Numerous conjectures are obtained through restricted iteration of an iterative algorithm, which is a widely used method in number theory, but people typically lack the means to demonstrate the accuracy and reasoned nature of conjectures. A fresh technique or new knowledge is frequently used to support a hypothesis. For the Collatz conjecture, we can describe it as a function:

The following sequence is obtained via the composite function (iteration): $a=\{n,T\left(n\right),T\left(T\left(n\right)\right),T\left(T\left(T\left(n\right)\right)\right),\cdots \}=\{n,T\left(n\right),T^2\left(n\right),T^3\left(n\right),\cdots \}$. Consequently, the Collatz conjecture can be stated as follows: The sequence a always leads to the integer 1, regardless of where you start with the natural number n, namely $T^m\left(n\right)=1$. The series a is an infinite sequence of ultimately period ${}^{[9,10]}$: the preperiod $\eta \left(n\right)$ varies with the initial value n, but the ultimately period is always $\{4,2,1\}$.

If a natural number can be divided by 2, it is said to be an even number; otherwise, it is said to be odd number. The Peano’s Axiom states that 1 is the smallest natural number. The set of natural numbers $N=\{1,2,3,\cdots \}$ may be separated into odd and even sets, we will utilize the standard definition of natural numbers in this work.

$\left\{\mathit{natural}\mathit{number}\right\}=\left\{\mathit{odd}\mathit{number}\right\}\bigcup \left\{\mathit{even}\mathit{number}\right\}$.

In the set of natural numbers where 1 is the smallest odd number and 2 is the smallest even number, we can use the expression $n=2k-1$ to indicate that it is an odd, and the expression $n=2k$ to indicate that it is an even, where k is any natural number.

We introduce two functions $O\left(x\right)=2x+1$ to express odd numbers greater than 1, and $E\left(x\right)=2x$ to express even numbers.

Definition 1 A natural number n is obtained by composition of the odd function $O\left(x\right)=2x+1$ and the even function $E\left(x\right)=2x$ several times, namely $n=f\left(f(\cdots f\left(1\right))\right)=f^k\left(1\right)$, the function f is either odd function $O\left(x\right)$ or even function $E\left(x\right)$.

For example, $7=2\ast 3+1=2\ast (2\ast 1+1+1)=O\left(O\right(1\left)\right),189=2\ast 2\ast (2\ast 4+1)=2\ast (2\ast (2\ast (2\ast 1))+1)=E\left(O\right(E\left(E\right(1\left)\right)\left)\right)$. In order to more clearly express the odd-even composition process of a natural number, we use binary representation of a natural number n, for example: $7={\left(111\right)}_2$, $18={\left(10010\right)}_2$.

Similarly, we can have a piecewise function

If $n=f\left(f(\cdots f\left(1\right))\right)=f^k\left(1\right)$, then the inverse function is $f^{-k}\left(n\right)=1$.

For generalization, we give the definitions of three natural numbers:

Definition 2(i) By applying the odd function $O\left(x\right)$m compositions, a natural number, namely $O^m\left(1\right)=2^m-1=2^{m-1}+2^{m-2}+\cdots +2+1={(11\cdots 1)}_2$, is obtained. such as $3={\left(11\right)}_2,7={\left(111\right)}_2,15={\left(1111\right)}_2,31={\left(11111\right)}_2,63={\left(111111\right)}_2,\cdots$, which we call it as pure odd number;

(ii) By applying the even function $E\left(x\right)$m compositions, a natural number, namely $E^m\left(1\right)=2^m={(10\cdots 0)}_2$, is obtained, such as, $2={\left(10\right)}_2,4={\left(100\right)}_2,8={\left(1000\right)}_2,16={\left(10000\right)}_2,32={\left(100000\right)}_2,64={\left(1000000\right)}_2,\cdots$, which we call it as pure even number;

(iii) The natural number obtained by the composition of odd function $O\left(x\right)$ and even function $E\left(x\right)$, we call it mixed number. Such as, $18={\left(10010\right)}_2,28={\left(11100\right)}_2,67={\left(1000011\right)}_2,309={\left(100110101\right)}_2$.

In particular, the natural numbers obtained by the finite alternately composition of the odd function $O\left(x\right)$ and the even function $E\left(x\right)$, namely, ${\left[E\left(O\left(1\right)\right)\right]}^m={(101\cdots 101)}_2$. Such as $5={\left(101\right)}_2,21={\left(10101\right)}_2,85={\left(1010101\right)}_2,341={\left(101010101\right)}_2,1365={\left(10101010101\right)}_2,5461={\left(1010101010101\right)}_2,\cdots$, which we call hard number.

Definition 3 The binary string of a natural number is a representation of its odd-even composite function, where the 1 in the $i(i>0)$-bit from right to left is the $i(i>0)$ sub-odd function, and 0 is the corresponding sub-even function.

For a natural number n, if its binary string has k bits, then the degree of composite function is $k-1$. The binary string of a pure odd number is made of all 1; and the binary string of a pure even number is all 0 aside from one 1 in left; the binary string of a mixed number is made of many 0 and 1. Figure 1 shows the decomposition of the composite function (inverse of the composite function) of 60.

In this way, we can classify the set of natural numbers in another way,

Property 4 The set of natural numbers can be divided into three different sets:

The pure odd number is an odd number, the pure even number is an even number, and mixed numbers can be either an odd number or an even number when compared to the conventional classification. The last bit of the binary string, which is either 0 or 1, indicates whether it is even or odd. The entire binary string implicitly indicates one of the following three types: all bits are 0s with the exception of one 1 in the left implicitly pure even, all bits are 1s implicitly pure odd, and there are 0 and 1 in any number of bits of the binary string implicitly mixed.

Example 1 (1)60, 97 are mixed numbers.

(2)64,1180591620717411303424 are pure even numbers.

(3)63,1180591620717411303423 are pure odd numbers.

${\mathbf{60}=\mathbf{2}\ast \mathbf{30}=\mathbf{2}\ast (\mathbf{2}\ast \mathbf{15})=\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast \mathbf{7}+\mathbf{1})))=\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast \mathbf{3})+\mathbf{1})+\mathbf{1})))=\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast \mathbf{1}+\mathbf{1})+\mathbf{1})+\mathbf{1})))=\mathit{E}(\mathit{E}\left(\mathit{O}\left(\mathit{O}\left(\mathit{O}\left(\mathbf{1}\right)\right)\right)\right)=\left(\mathbf{111100}\right)}_{\mathbf{2}}$;

${\mathbf{97}=\mathbf{2}\ast \mathbf{48}+\mathbf{1}=\mathbf{2}\ast (\mathbf{2}\ast \mathbf{24})+\mathbf{1}=\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast \mathbf{12}))+\mathbf{1}=\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast \mathbf{6})))+\mathbf{1}=\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast \mathbf{3})))+\mathbf{1}=\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast (\mathbf{2}\ast \mathbf{1}+\mathbf{1}))))+\mathbf{1}=\mathit{O}\left(\mathit{E}\left(\mathit{E}\left(\mathit{E}\left(\mathit{E}\left(\mathit{O}\left(\mathbf{1}\right)\right)\right)\right)\right)\right)=\left(\mathbf{1100001}\right)}_{\mathbf{2}}$.

$\mathbf{64}={\mathbf{2}}^{\mathbf{6}}={\left(\mathbf{1000000}\right)}_{\mathbf{2}},\mathbf{1180591620717411303424}={\mathbf{2}}^{\mathbf{70}}=(\mathbf{10000}\cdots \mathbf{0})$ are pure even numbers:

$\mathbf{63}={\mathbf{2}}^{\mathbf{6}}-\mathbf{1}={\left(\mathbf{111111}\right)}_{\mathbf{2}},\mathbf{1180591620717411303423}={\mathbf{2}}^{\mathbf{70}}-\mathbf{1}={(\mathbf{11}\cdots \mathbf{1})}_{\mathbf{2}}$ are pure odd numbers.

This composite function has special significance, for example,

Example 2 Let $\left[\mathit{x}\right]$ be $\mathit{x}$ the integer part, that is, $\left[\mathit{x}\right]=\mathit{max}\{\mathit{y}\in \mathit{Z}:\mathit{y}\le \mathit{x}\}$, $\left\{\mathit{x}\right\}$ be fractional part, namely $\left\{\mathit{x}\right\}=\mathit{x}-\left[\mathit{x}\right]$. Assuming $\mathit{g}\left(\mathit{x}\right)=\frac{\mathbf{1}}{\left[\mathit{x}\right]-\left\{\mathit{x}\right\}+\mathbf{1}}$, is rational set ${\mathit{Q}}^{+}$ and rational set $\mathit{Q}$ can be expressed as respectively

${\mathit{Q}}^{+}=\{\mathit{g}\left(\mathbf{0}\right),{\mathit{g}}^{\mathbf{2}}\left(\mathbf{0}\right),{\mathit{g}}^{\mathbf{3}}\left(\mathbf{0}\right),\cdots \}$,

$\mathit{Q}=\{\mathbf{0},\mathit{g}\left(\mathbf{0}\right),-\mathit{g}\left(\mathbf{0}\right),{\mathit{g}}^{\mathbf{2}}\left(\mathbf{0}\right),-{\mathit{g}}^{\mathbf{2}}\left(\mathbf{0}\right),{\mathit{g}}^{\mathbf{3}}\left(\mathbf{0}\right),-{\mathit{g}}^{\mathbf{3}}\left(\mathbf{0}\right),\cdots \}$ .

See the references${}^{\left[\mathbf{5}\right]}$ for proof.

For the Collatz conjecture, if expressed by the function $\mathit{T}\left(\mathit{n}\right)$, we find that ${\mathit{T}}^{\mathit{k}}\left(\mathit{n}\right)=\mathbf{1}$.

For the sake of discussion, we combine the compound function (iterative relation) to get the reduced Collatz function

The result is an odd number. We introduce tabular form and algebraic expression to express the reduced Collatz function $\mathit{RT}\left(\mathit{n}\right)$.

For example, for $\mathit{n}=\mathbf{67}$ we use the formula (1.1) and iteration, get the following table. Where the last column is its algebraic expression:.

67→	202 →	101			$\frac{{\mathbf{3}}^{\mathbf{8}}}{{\mathbf{2}}^{\mathbf{19}}}·\mathbf{67}$	$\frac{{\mathbf{3}}^{\mathbf{7}}}{{\mathbf{2}}^{\mathbf{19}}}$
101→	304→	152→	76→	38→	19	$\frac{{\mathbf{3}}^{\mathbf{6}}}{{\mathbf{2}}^{\mathbf{18}}}$
19→	58→	29				$\frac{{\mathbf{3}}^{\mathbf{5}}}{{\mathbf{2}}^{\mathbf{14}}}$
29→	88→	44→	22→	11		$\frac{{\mathbf{3}}^{\mathbf{4}}}{{\mathbf{2}}^{\mathbf{13}}}$
11→	34→	17				$\frac{{\mathbf{3}}^{\mathbf{3}}}{{\mathbf{2}}^{\mathbf{10}}}$
17→	52→	26→	13			$\frac{{\mathbf{3}}^{\mathbf{2}}}{{\mathbf{2}}^{\mathbf{9}}}$
13→	40→	20→	10→	5		$\frac{\mathbf{3}}{{\mathbf{2}}^{\mathbf{7}}}$
5→	16→	8→	4→	2→	1	$\frac{\mathbf{1}}{{\mathbf{2}}^{\mathbf{4}}}$
1

Representing the numbers in the table by its binary string to get

1000011→	11001010→	1100101			$\frac{{\mathbf{3}}^{\mathbf{8}}}{{\mathbf{2}}^{\mathbf{19}}}·\mathbf{67}$	$\frac{{\mathbf{3}}^{\mathbf{7}}}{{\mathbf{2}}^{\mathbf{19}}}$
1100101→	100110000→	10011000→	1001100→	100110→	10011	$\frac{{\mathbf{3}}^{\mathbf{6}}}{{\mathbf{2}}^{\mathbf{18}}}$
10011→	111010→	11101				$\frac{{\mathbf{3}}^{\mathbf{5}}}{{\mathbf{2}}^{\mathbf{14}}}$
11101→	1011000→	101100→	10110→	1011		$\frac{{\mathbf{3}}^{\mathbf{4}}}{{\mathbf{2}}^{\mathbf{13}}}$
1011→	100010→	10001				$\frac{{\mathbf{3}}^{\mathbf{3}}}{{\mathbf{2}}^{\mathbf{10}}}$
10001→	110100→	11010→	1101			$\frac{{\mathbf{3}}^{\mathbf{2}}}{{\mathbf{2}}^{\mathbf{9}}}$
1101→	101000→	10100→	1010→	101		$\frac{\mathbf{3}}{{\mathbf{2}}^{\mathbf{7}}}$
101→	10000→	1000→	100→	10→	1	$\frac{\mathbf{1}}{{\mathbf{2}}^{\mathbf{4}}}$
1

The above table is simplified with the help of the reduced Collatz function, and the simplified binary string table obtained is as follows, and we will use this form as the default table in the rest of this article.

1000011→	11001010→	1100101	$\frac{{\mathbf{3}}^{\mathbf{8}}}{{\mathbf{2}}^{\mathbf{19}}}·\mathbf{67}$	$\frac{{\mathbf{3}}^{\mathbf{7}}}{{\mathbf{2}}^{\mathbf{19}}}$
1100101→	100110000→	10011		$\frac{{\mathbf{3}}^{\mathbf{6}}}{{\mathbf{2}}^{\mathbf{18}}}$
10011→	111010→	11101		$\frac{{\mathbf{3}}^{\mathbf{5}}}{{\mathbf{2}}^{\mathbf{14}}}$
11101→	1011000→	1011		$\frac{{\mathbf{3}}^{\mathbf{4}}}{{\mathbf{2}}^{\mathbf{13}}}$
1011→	100010→	10001		$\frac{{\mathbf{3}}^{\mathbf{3}}}{{\mathbf{2}}^{\mathbf{10}}}$
10001→	110100→	1101		$\frac{{\mathbf{3}}^{\mathbf{2}}}{{\mathbf{2}}^{\mathbf{9}}}$
1101→	101000→	101		$\frac{\mathbf{3}}{{\mathbf{2}}^{\mathbf{7}}}$
101→	10000→	1		$\frac{\mathbf{1}}{{\mathbf{2}}^{\mathbf{4}}}$
1

In the future, we will use this table as a research tool to write the algebraic expression of the last column in the table:

In general, the algebraic expressions are obtained: Starting from the last row of the table and going up to the binary corresponding to the initial value n of the first row, the numerator is ${\mathbf{3}}^{\mathit{k}},\mathit{k}=\mathbf{0},\mathbf{1},\mathbf{2},\mathbf{3},\cdots$, The denominator is ${\mathbf{2}}^{{\mathit{r}}_{\mathit{k}}}$, ${\mathit{r}}_{\mathbf{0}}={\mathit{m}}_{\mathbf{0}},{\mathit{r}}_{\mathit{k}}={\mathit{r}}_{\mathit{k}-\mathbf{1}}+{\mathit{m}}_{\mathit{k}},\mathit{k}=\mathbf{1},\mathbf{2},\mathbf{3},\cdots$, ${\mathit{m}}_{\mathit{k}}$ here for the before k lines at the end of the second column of binary string number 0. The details are expressed in the last line of the corresponding table, and we write out the algebraic expression of them as follows:

The Collatz function is expressed in binary form as

The characteristics of the first and last parts are represented by the Figure 2.

By observing the binary strings of iterated Collatz functions with initial values of 31,63 (see the tabulation procedure in reference [7]) and 97,10027 as the follows, we get some laws of pure even, pure odd and mixed numbers with initial values. In the following, we will analyze and discuss the table of the iterative process of the Collatz functions from the two dimensions of row and column of the tables.

			$\frac{{\mathbf{3}}^{\mathbf{43}}}{{\mathbf{2}}^{\mathbf{75}}}·\mathbf{97}$
$\mathbf{97}={\left(\mathbf{1100001}\right)}_{\mathbf{2}}$→	${\left(\mathbf{100100100}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1001001}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{42}}}{{\mathbf{2}}^{\mathbf{75}}}$
$\mathbf{73}={\left(\mathbf{1001001}\right)}_{\mathbf{2}}$→	${\left(\mathbf{11011100}\right)}_{\mathbf{2}}$→	${\left(\mathbf{110111}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{41}}}{{\mathbf{2}}^{\mathbf{73}}}$
$\mathbf{55}={\left(\mathbf{110111}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10100110}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1010011}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{40}}}{{\mathbf{2}}^{\mathbf{71}}}$
$\mathbf{83}={\left(\mathbf{1010011}\right)}_{\mathbf{2}}$→	${\left(\mathbf{11111010}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1111101}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{39}}}{{\mathbf{2}}^{\mathbf{70}}}$
$\mathbf{125}={\left(\mathbf{1111101}\right)}_{\mathbf{2}}$→	${\left(\mathbf{101111000}\right)}_{\mathbf{2}}$→	${\left(\mathbf{101111}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{38}}}{{\mathbf{2}}^{\mathbf{69}}}$
$\mathbf{47}={\left(\mathbf{101111}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10001110}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1000111}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{37}}}{{\mathbf{2}}^{\mathbf{66}}}$
$\mathbf{71}={\left(\mathbf{1000111}\right)}_{\mathbf{2}}$→	${\left(\mathbf{11010110}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1101011}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{36}}}{{\mathbf{2}}^{\mathbf{65}}}$
$\mathbf{107}={\left(\mathbf{1101011}\right)}_{\mathbf{2}}$→	${\left(\mathbf{101000010}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10100001}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{35}}}{{\mathbf{2}}^{\mathbf{64}}}$
$\mathbf{161}={\left(\mathbf{10100001}\right)}_{\mathbf{2}}$→	${\left(\mathbf{111100100}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1111001}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{34}}}{{\mathbf{2}}^{\mathbf{63}}}$
$\mathbf{121}={\left(\mathbf{1111001}\right)}_{\mathbf{2}}$→	${\left(\mathbf{101101100}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1011011}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{33}}}{{\mathbf{2}}^{\mathbf{61}}}$
$\mathbf{91}={\left(\mathbf{1011011}\right)}_{\mathbf{2}}$→	${\left(\mathbf{100010010}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10001001}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{32}}}{{\mathbf{2}}^{\mathbf{59}}}$
$\mathbf{137}={\left(\mathbf{10001001}\right)}_{\mathbf{2}}$→	${\left(\mathbf{110011100}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1100111}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{31}}}{{\mathbf{2}}^{\mathbf{58}}}$
$\mathbf{103}={\left(\mathbf{1100111}\right)}_{\mathbf{2}}$→	${\left(\mathbf{100110110}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10011011}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{30}}}{{\mathbf{2}}^{\mathbf{56}}}$
$\mathbf{155}={\left(\mathbf{10011011}\right)}_{\mathbf{2}}$→	${\left(\mathbf{111010010}\right)}_{\mathbf{2}}$→	${\left(\mathbf{11101001}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{29}}}{{\mathbf{2}}^{\mathbf{55}}}$
$\mathbf{233}={\left(\mathbf{11101001}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1010111100}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10101111}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{28}}}{{\mathbf{2}}^{\mathbf{54}}}$
$\mathbf{175}={\left(\mathbf{10101111}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1000001110}\right)}_{\mathbf{2}}$→	${\left(\mathbf{100000111}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{27}}}{{\mathbf{2}}^{\mathbf{52}}}$
$\mathbf{263}={\left(\mathbf{100000111}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1100010110}\right)}_{\mathbf{2}}$→	${\left(\mathbf{110001011}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{26}}}{{\mathbf{2}}^{\mathbf{51}}}$
$\mathbf{395}={\left(\mathbf{110001011}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10010100010}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1001010001}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{25}}}{{\mathbf{2}}^{\mathbf{50}}}$
$\mathbf{593}={\left(\mathbf{1001010001}\right)}_{\mathbf{2}}$→	${\left(\mathbf{11011110100}\right)}_{\mathbf{2}}$→	${\left(\mathbf{110111101}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{24}}}{{\mathbf{2}}^{\mathbf{49}}}$
$\mathbf{445}={\left(\mathbf{110111101}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10100111000}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10100111}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{23}}}{{\mathbf{2}}^{\mathbf{47}}}$
$\mathbf{167}={\left(\mathbf{10100111}\right)}_{\mathbf{2}}$→	${\left(\mathbf{111110110}\right)}_{\mathbf{2}}$→	${\left(\mathbf{11111011}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{22}}}{{\mathbf{2}}^{\mathbf{44}}}$
$\mathbf{251}={\left(\mathbf{11111011}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1011110010}\right)}_{\mathbf{2}}$→	${\left(\mathbf{101111001}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{21}}}{{\mathbf{2}}^{\mathbf{43}}}$
$\mathbf{377}={\left(\mathbf{101111001}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10001101100}\right)}_{\mathbf{2}}$→	${\left(\mathbf{100011011}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{20}}}{{\mathbf{2}}^{\mathbf{42}}}$
$\mathbf{283}={\left(\mathbf{100011011}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1101010010}\right)}_{\mathbf{2}}$→	${\left(\mathbf{110101001}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{19}}}{{\mathbf{2}}^{\mathbf{40}}}$
$\mathbf{425}={\left(\mathbf{110101001}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10011111100}\right)}_{\mathbf{2}}$→	${\left(\mathbf{100111111}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{18}}}{{\mathbf{2}}^{\mathbf{39}}}$
$\mathbf{319}={\left(\mathbf{100111111}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1110111110}\right)}_{\mathbf{2}}$→	${\left(\mathbf{111011111}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{17}}}{{\mathbf{2}}^{\mathbf{37}}}$
$\mathbf{479}={\left(\mathbf{111011111}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10110011110}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1011001111}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{16}}}{{\mathbf{2}}^{\mathbf{36}}}$
$\mathbf{719}={\left(\mathbf{1011001111}\right)}_{\mathbf{2}}$→	${\left(\mathbf{100001101110}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10000110111}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{15}}}{{\mathbf{2}}^{\mathbf{35}}}$
$\mathbf{1079}={\left(\mathbf{10000110111}\right)}_{\mathbf{2}}$→	${\left(\mathbf{110010100110}\right)}_{\mathbf{2}}$→	${\left(\mathbf{11001010011}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{14}}}{{\mathbf{2}}^{\mathbf{34}}}$
$\mathbf{1619}={\left(\mathbf{11001010011}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1001011111010}\right)}_{\mathbf{2}}$→	${\left(\mathbf{100101111101}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{13}}}{{\mathbf{2}}^{\mathbf{33}}}$
$\mathbf{2429}={\left(\mathbf{100101111101}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1110001111000}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1110001111}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{12}}}{{\mathbf{2}}^{\mathbf{32}}}$
$\mathbf{911}={\left(\mathbf{1110001111}\right)}_{\mathbf{2}}$→	${\left(\mathbf{101010101110}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10101010111}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{11}}}{{\mathbf{2}}^{\mathbf{29}}}$
$\mathbf{1367}={\left(\mathbf{10101010111}\right)}_{\mathbf{2}}$	${\left(\mathbf{1000000000110}\right)}_{\mathbf{2}}$	${\left(\mathbf{100000000011}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{10}}}{{\mathbf{2}}^{\mathbf{28}}}$
$\mathbf{2051}={\left(\mathbf{100000000011}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1100000001010}\right)}_{\mathbf{2}}$→	${\left(\mathbf{110000000101}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{9}}}{{\mathbf{2}}^{\mathbf{27}}}$
$\mathbf{3077}={\left(\mathbf{110000000101}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10010000010000}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1001000001}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{8}}}{{\mathbf{2}}^{\mathbf{26}}}$
$\mathbf{577}={\left(\mathbf{1001000001}\right)}_{\mathbf{2}}$→	${\left(\mathbf{11011000100}\right)}_{\mathbf{2}}$→	${\left(\mathbf{110110001}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{7}}}{{\mathbf{2}}^{\mathbf{22}}}$
$\mathbf{433}={\left(\mathbf{110110001}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10100010100}\right)}_{\mathbf{2}}$→	${\left(\mathbf{101000101}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{6}}}{{\mathbf{2}}^{\mathbf{20}}}$
$\mathbf{325}={\left(\mathbf{101000101}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1111010000}\right)}_{\mathbf{2}}$→	${\left(\mathbf{111101}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{5}}}{{\mathbf{2}}^{\mathbf{18}}}$
$\mathbf{61}={\left(\mathbf{111101}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10111000}\right)}_{\mathbf{2}}$→	${\left(\mathbf{10111}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{4}}}{{\mathbf{2}}^{\mathbf{14}}}$
$\mathbf{23}={\left(\mathbf{10111}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1000110}\right)}_{\mathbf{2}}$→	${\left(\mathbf{100011}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{3}}}{{\mathbf{2}}^{\mathbf{11}}}$
$\mathbf{35}={\left(\mathbf{100011}\right)}_{\mathbf{2}}$→	${\left(\mathbf{1101010}\right)}_{\mathbf{2}}$→	${\left(\mathbf{110101}\right)}_{\mathbf{2}}$	$\frac{{\mathbf{3}}^{\mathbf{2}}}{{\mathbf{2}}^{\mathbf{10}}}$
$\mathbf{53}={\left(\mathbf{110101}\right)}_{\mathbf{2}}$	${\left(\mathbf{10100000}\right)}_{\mathbf{2}}$	${\left(\mathbf{101}\right)}_{\mathbf{2}}$	$\frac{\mathbf{3}}{{\mathbf{2}}^{\mathbf{9}}}$
$\mathbf{5}={\left(\mathbf{101}\right)}_{\mathbf{2}}$	${\left(\mathbf{10000}\right)}_{\mathbf{2}}$	${\left(\mathbf{1}\right)}_{\mathbf{2}}$	$\frac{\mathbf{1}}{{\mathbf{2}}^{\mathbf{4}}}$

Comparing the Collatz function $\mathit{T}\left(\mathit{x}\right)$ and the function ${\mathit{f}}^{-\mathbf{1}}\left(\mathit{x}\right)$, if their domain is defined as the set of natural numbers, we find that they have the following relation:

1) The function ${\mathit{f}}^{-\mathbf{1}}\left(\mathit{x}\right)$ is strictly monotonically decreasing,

2) When $\mathit{x}$ is purely even, the function $\mathit{T}\left(\mathit{x}\right)$ is only one case of the functions, is strictly monotonically decreasing;

3) When $\mathit{x}$ is a pure or mixed odd number, the function $\mathit{T}\left(\mathit{x}\right)$ is wavy, which is increasing, followed by one or more decreasing processes, that is, "increase – decrease – increase", or "increase – decrease ⋯ decrease – increase". For example, Figure 3 and Figure 4 are the plots of the iterated sequence of Collatz functions with initial values of pure odd $\mathbf{255}={\mathbf{2}}^{\mathbf{8}}-\mathbf{1}={\left(\mathbf{11111111}\right)}_{\mathbf{2}}$ and mixed odd number $\mathbf{97}={\left(\mathbf{1100001}\right)}_{\mathbf{2}}$, respectively.

Due to the fact that an odd number can be either pure or mixed, when $\mathit{x}$ is odd, the Collatz function $\mathbf{3}\mathit{x}+\mathbf{1}$ can be parted into two parts. i.e., $\mathbf{3}\mathit{x}+\mathbf{1}=\mathbf{2}\mathit{x}+(\mathit{x}+\mathbf{1})$.