Write a function that takes a positive integer n as an input and performs the following steps. If n is even, divide it by 2, if n is odd, multiply it by 3 and add 1. Repeat the process (which has been called "Half Or Triple Plus One") indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. What is the smallest number that has not been proven to reach 1?

The Collatz Conjecture, also known as the 3n+1 conjecture, is an unsolved problem in mathematics. Despite extensive computational evidence supporting the conjecture, no general proof has been found, and it remains an open question. Therefore, there is no known smallest number for which the conjecture hasn’t been proven to reach 1. Nonetheless, for all numbers tested up to this point (including numbers as large as 2⁶⁰), the conjecture holds.

Here is a simple function in Python to perform the "Half Or Triple Plus One" operations:

def collatz(n):
    if n <= 0:
        raise ValueError("Input must be a positive integer.")
    
    while n != 1:
        print(n)
        if n % 2 == 0:
            n = n // 2
        else:
            n = 3 * n + 1
    print(n)

This function takes a positive integer n as input and performs the Half Or Triple Plus One algorithm described in the Collatz conjecture. As a result, it prints the sequence of numbers visited during this process, including the final value of 1.

Keep in mind that without a rigorous proof that confirms the validity of the conjecture, it’s impossible to determine the smallest integer for which the conjecture fails.