Collatzlike

Collatzlike refers to a broad class of integer dynamical systems that generalize the structure of the Collatz (3n+1) problem. In its typical form, a Collatzlike map f operates on positive integers by applying a parity-dependent rule, most often f(n) = n/2 when n is even and f(n) = a n + b when n is odd, where a and b are fixed integers chosen to create a nontrivial, integer-valued trajectory. The original Collatz map corresponds to a = 3 and b = 1.

Variants expand or modify the rule. Some replace the odd-step with other linear forms, such as f(n) = a n + b for odd n with different (a,b); others adjust the even-step or incorporate more residue classes. Together these define numerous Collatzlike problems, all sharing the same basic dynamical structure: iterating a simple rule on integers and studying the long-term behavior of trajectories.

Research questions focus on convergence, cycles, and stopping times. A central question is whether every starting value eventually enters a finite cycle (in the classic case, the cycle 1 → 4 → 2 → 1), or whether some trajectories diverge. For many parameter choices, the global behavior is unknown; some parameter sets are provably divergent or eventually periodic, while many remain open and are explored computationally.

The term Collatzlike is used across number theory and dynamical systems to denote problems that resemble the original conjecture in structure, regardless of the specific constants. The study connects with generalized Collatz problems, total stopping time, and analysis of integer maps and cycles.