The Collatz Conjecture remains an unsolved problem in number theory, which posits that iteratively applying a simple function to any starting odd number will eventually lead to the number 1. In this paper, we investigate the behavior of the Collatz function for odd numbers with different prime factorizations using a combination of number theory, algebraic properties, and computational techniques. Our study reveals new insights into the role of prime factors and topological properties in the behavior of the Collatz function. We define the subspace H3, consisting of numbers divisible by 3, and analyze its topological properties, finding that its fundamental group is non-trivial. This suggests non-trivial behavior in the Collatz sequences for numbers in H3. We observe an intersection between the Collatz sequences for odd numbers and numbers in H3, indicating that the sequences for odd numbers may eventually reach a number in H3, warranting further investigation. We compare the properties of sequences that intersect with H3 and those that do not, finding significant differences in sequence length, maximum value, and convergence rate. Additionally, we generate prime factorizations for each number in the sequences and analyze the distribution of primes within the sequences. Our analysis reveals correlations between the number of prime factors, the largest prime factor, and sequence properties such as average length and maximum value. These findings offer new insights into the behavior of the Collatz function for odd numbers, contributing to a deeper understanding of the Collatz Conjecture and its underlying structure. The results open up potential avenues for further research, including exploring number-theoretic properties, algebraic properties, asymptotic analysis, or connections to other problems in number theory or computer science.
