$3n + 3^k$ Problem

TL;DR

This paper generalizes the Collatz problem to the $3n + 3^k$ problem, showing that its sequences converge to a cycle passing through $3^k$, and relates its iterates to those of the Collatz function.

Contribution

It introduces a new generalized problem, the $3n + 3^k$ problem, and proves its convergence properties based on the Collatz problem.

Findings

Sequences converge to cycles passing through $3^k$

The $3n + 3^k$ sequence relates directly to Collatz iterates for scaled inputs

Convergence behavior mirrors that of the Collatz problem

Abstract

The Collatz problem is generalized into $3n + 3^k$ problem. It is shown that as long as the Collatz function iterates converge to the cycle passing through the number 1, the $3n + 3^k$ sequence converges to the cycle passing through the number $3^k$ for arbitrary positive integers $n$ and $k$. The proof shows that the sequence of $3n + 3^k$ function iterates for a number $3^k n$ is exactly the sequence of the Collatz function iterates for $n$ multiplied by $3^k$.