Posts

As part of personal studies, I was leaning how to build R packages and deploy through Github. Essentially, as a practice exercise, I wanted to model the famous Collatz conjucture and visualize the plot for various random numbers.

The problem

Consider an arbitrary positive integer n:

• If n is even, divide it by 2.
• Else, if n is odd, multiply it by 3 and add 1 to the product.

Therefore, we can define a function fⁱ(n) such that:

\( f(n) = \begin{cases} n/2 & \text{if } n \equiv 0 \mod 2, \\ 3n+1 & \text{if } n \equiv 1 \mod 2. \end{cases} \)

Subsequently, we need to loop the above equation sequencially by taking the result at the previous step as the starting value at the next such that:

\( k_i = \begin{cases} n & \text{for } i = 0, \\ f(k_{i-1}) & \text{for } i > 0 \end{cases} \)

The conjecture states that this process will eventually reach the number 1, regardless of which arbitrary positive integer is chosen as the starting (initial) value n. Once the process reaches unity, a perpetual loop is reached: 4,3,2,1,4,3,2,1... Therefore, we stop the loop once we reach 1.

Currently, the highest arbitrary integer that has been officially proven to concur with this behaviour is 2^100,000 - 1. Can this 3n + 1 hold for bigger integers beyond this?

Installing the CollatzR package into R

To install the CollatzR package into R, you can follow the following procedure:

1. Ensure you are connected to the internet.
2. Open your R software and open a new Rscript. Install devtools package into your library. This can be done by typing (or copy-pasting) this code into your R script and running it:

   
               install.packages("devtools")# you need devtools package for the installation of packages from Github
               library(devtools)#load the devtools package ito your library

3. Use devtools to install the CollatzR package into R.

   
               install_github("SulmanOlieko/collatzR")#Install CollatzR
               library(collatzR)#load the CollatzR package into R

Usage

Generate hailstone sequences

You can now generate the hailstone sequences of any arbitrary integer between 1 - 2¹⁰²⁴. To do this, simply type collatz_sequence() and input your integer inside the parentheses and run it. For instance, if I want to see the Collatz sequence for 123456789, i would:

collatz_sequence(3732423)

Visualize the plot

To view the plot for this number:

 visualize_collatz(3732423)

How many steps or stopping times are there for this number before reaching 1?

       collatz(3732423)
       #alternatively:
       collatz_sequence(3732423)$step

Identify the maximum value in the sequence:

        max_collatz(3732423)
        #alternatively:
        collatz_sequence(3732423)$max_value

Other operations

The CollatzR package can also perform operations such as:

1. Determining if a number is odd or even:

   is.even(3732423)

2. Determining if a number is whole:

   is.wholenumber(3732423)

Happy interacting with the tool!