Classification of Integers and Occurrence Patterns of Multiples of 3 in the Inverse Collatz Map

AbstractWe study the inverse map associated with the Collatz map, focusing on theconditions under which an inverse iterate is an integer and under which thatinteger is divisible by 3. We organize the exponent conditions according to aclassification of integers and use the periodicity of powers of 2 to examine theoccurrence of multiples of 3. Based on these results, we discuss the possibilityof cycles in the inverse map and state conjectures for future investigation.