Operation Table

From natural numbers to real numbers.

There is also an alternative approach to analyzing trajectories which we refer to as the operational approach. Recall the functions $R$ and $R_0$ from the Introduction chapter.

These were defined for the natural number domain and odd number range. Now, we will introduce their brute versions, which take a real number input instead and output a real number. The $x$ parameter that was well-defined in the prior functions, is now given explicitly; and it does not respect whether the resulting number is an integer or not; hence the name "brute".

• Brute Reduced Collatz function $\mathcal{R}:\mathbb{R} \times \mathbb{N} \to \mathbb{R}$ is defined as:

$$\mathcal{R}(n, x) = \frac{3n+1}{2^{x}}$$

• Brute Reduced Collatz Initializer function $\mathcal{R}_0 : \mathbb{R} \times \mathbb{N} \to \mathbb{R}$ is defined as:

$$\mathcal{R}_0(n, x) = \frac{n}{2^x}$$

Since $x$ is given explicitly, we can give "wrong" parameters and end up with non-integer results. That is what makes these functions imprecise.

Imprecise Trajectories

For an ECF $\{p_0, p_1, \ldots, p_m\}$, consider the imprecise trajectory starting from some number $n$:

$$n \xrightarrow{p_0} \mathcal{R}_0(n, p_0) \xrightarrow{p_1 - p_0} \mathcal{R}(n_0, p_1 - p_0) \xrightarrow{p_2 - p_1} \ldots \xrightarrow{p_m - p_{m-1}} \mathcal{R}(n_{m-1}, p_m - p_{m-1})$$

The notation is similar to what we have done in the Trajectory chapter:

• $n_0 = \mathcal{R}_0(n, p_0)$
• $n_i = \mathcal{R}(n_{i-1}, p_i - p_{i-1})$

ICF is defined exactly the same way for imprecise trajectories too! This makes it a really powerful tool, which we will soon make use of.

Brute Operation

Define $\beta : \mathbb{N} \times \mathbb{N} \to \mathbb{R}$ as follows:

• Consider an ECF $\{p_0, p_1, \ldots, p_m\}$, and its bijective map that is $\rho = 2^{p_0} + 2^{p_1} + \ldots + 2^{p_m}$.
• Consider the imprecise trajectory that is defined by this ECF and the starting number is $n$:

$$n \xrightarrow{p_0} \mathcal{R}_0(n, p_0) \xrightarrow{p_1 - p_0} \mathcal{R}(n_0, p_1 - p_0) \xrightarrow{p_2 - p_1} \ldots \xrightarrow{p_m - p_{m-1}} \mathcal{R}(n_{m-1}, p_m - p_{m-1})$$

• As per our definition, the terminating number here is denoted as $n_m$.
• The function is then defined as $\beta(n, \rho) = n_m$.

Here is how the first 10 numbers look like in the operation table for $\beta$:

β      1    2     3     4     5     6      7      8      9     10
1    1.0* 0.5   2.0  0.25  1.00* 1.25   3.50  0.125  0.500  0.625
2    2.0  1.0*  3.5  0.50  1.75  2.00   5.75  0.250  0.875  1.000*
3    3.0  1.5   5.0  0.75  2.50  2.75   8.00  0.375  1.250  1.375
4    4.0  2.0   6.5  1.00* 3.25  3.50  10.25  0.500  1.625  1.750
5    5.0  2.5   8.0  1.25  4.00  4.25  12.50  0.625  2.000  2.125
6    6.0  3.0   9.5  1.50  4.75  5.00  14.75  0.750  2.375  2.500
7    7.0  3.5  11.0  1.75  5.50  5.75  17.00  0.875  2.750  2.875
8    8.0  4.0  12.5  2.00  6.25  6.50  19.25  1.000* 3.125  3.250
9    9.0  4.5  14.0  2.25  7.00  7.25  21.50  1.125  3.500  3.625
10  10.0  5.0  15.5  2.50  7.75  8.00  23.75  1.250  3.875  4.000

If the cell contains 1, then the corresponding column $\rho$ actually represents the ECF that is of the Collatz sequence of $n$. That is awesome, because you can immediately learn the sequence just by looking at the table.

There are many ways we can analyze the table, see the following links:

Navigating

Testing an ECF

Shrinking Numbers