The Collatz Problem generalized to 3x+k

TL;DR

This paper investigates the generalized Collatz problem with 3x+k, providing limit cycles, probability relations, and confirming that Tao's oscillation result applies broadly for k not divisible by 2 or 3.

Contribution

It extends the analysis of the Collatz problem to the 3x+k case, establishing limit cycles and a probability relation, and verifies Tao's oscillation result for a wide range of k.

Findings

Limit cycles identified up to k=9997

Probability distribution relation for Syracuse iterates derived

Tao's oscillation result holds for all k not divisible by 2 or 3

Abstract

The Collatz problem with $3x+k$ is revisited. Positive and negative limit cycles are given up to k=9997 starting with $x_0=-2\cdot10^7...+2\cdot10^7$. A simple relation between the probability distribution for the Syracuse iterates for various k (not divisible by 2 and 3) is obtained. From this it follows that the oscillation considered by Tao 2019 ( arXiv:1909.03562v2 ) does not depend on k. Thus this piece of the proof of his theorem 1.3 "Almost all Collatz orbits attain almost bounded values" holds for all k not divisible by 2 and 3.