The Collatz conjecture, as known as the Hailstone sequence, is the 3n+1 problem which can be defined as the following function f(n) where n must be a positive integer greater than one and f(n) is used to produce the next value in the Collatz conjecture.

3 3 
'Please enter a positive integer.' 'Please enter a positive integer.' 
28 28 
To obain all iterations within a specific n value's Collatz conjecture, the equation f(n) is used continuously until terminated by the values 421. Sage is a very useful tool when computing conditonal values as such, especially larger values with lots of iterations.


[12, 6, 3, 10, 5, 16, 8, 4, 2, 1] [12, 6, 3, 10, 5, 16, 8, 4, 2, 1] 
[29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1] [29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1] 
[44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1] [44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1] 

What if we only want the value for the number of iterations it takes f(n) to reach the Collatz conjecture?

10 10 
25 25 
103 103 
19 19 
Now that we can calculate the value of f(n), the entire range of values for f(n), and the number of iterations of f(n), we will observe their behavior linearly.

Here we can see the hailstone values for n with a range up to 10,000. Although f(12) only has 10 iterations to reach the Collatz conjecture, we can observe all iterations of every value produced.



Notice, there are values far from the general trend of log and decay curves. These values are called champions, which take longer than any other iterations within f(n).

We can see that the champion values have a general pattern of the log curve. Why is this important? Taking this for example and looking back at the previous hailstone plots, we can see that 2 general patterns occur, a positve logrithmic curve a negative decay curve. What does tell us about the iterations of f(n)?
