The Collatz Conjecture, or \(3X + 1\) problem, is one of the most famous unsolved problems in mathematics. It concerns sequences of integers (also known as hailstone sequences) in which the next term, n, is obtained from the previous term, p, as follows: if p is even, \(n = \frac{p}{2}\) . If p is odd, \(n = 3p + 1\) . The conjecture is that these sequences always reach 1 (and fall into the 1, 4, 2 loop), no matter which positive integer is chosen to start the sequence. In this paper, an extension of this function is explored, and we discuss the cases in which the sequence diverges and loops, as well as go into further detail about the loops. The bulk of the paper deals with a function similar to that of the one used in the Collatz Conjecture, but instead of \(n = 3p + 1\) when p is odd, \(n = p + b\) , where b is a positive integer. This paper also explores the number and types of loops that can be formed, proving some properties of loops and sequences along the way, before eventually providing a general formula for the number of possible loops from any choice of b.