Generalized Collatz Sequences Based on Consecutive Positive Integer Pairs

During an investigation into the fundamental role of the consecutive integers 2 and 3 (division by 2 and multiplication by 3) in the Collatz map, several remarkable patterns related to powers of these two numbers were observed. These observations led to a deeper understanding of the structure of the Collatz map and naturally gave rise to a family of related mappings, which are presented in this paper. Based on these findings, this paper shows that the Collatz map can be reformulated for other pairs of consecutive integers, such as 3 and 4 (division by 3 and multiplication by 4), and consequently, the original Collatz map may naturally be regarded as the first member of this family of mappings.