The relationship between stopping time and number of odd terms in Collatz sequences

TL;DR

This paper introduces a numerical formula linking stopping time and odd terms in Collatz sequences, tested extensively and proposed as a conjecture to aid in proving the Collatz conjecture.

Contribution

It presents a new formula relating stopping time and odd terms in Collatz sequences, supported by extensive numerical testing up to large bounds.

Findings

Formula tested for all numbers up to 10^7

Supported by random tests up to 2^128

Proposed as a conjecture to assist in proving the Collatz conjecture

Abstract

The Collatz sequence for a given natural number $N$ is generated by repeatedly applying the map $N$ $\rightarrow$ $3N+1$ if $N$ is odd and $N$ $\rightarrow$ $N/2$ if $N$ is even. One elusive open problem in Mathematics is whether all such sequences end in 1 (Collatz conjecture), the alternative being the possibility of cycles or of unbounded sequences. In this paper, we present a formula relating the stopping time and the number of odd terms in a Collatz sequence, obtained numerically and tested for all numbers up to $10^7$ and for random numbers up to $2^{128.000}$. This result is presented as a conjecture, and with the hope that it could be useful for constructing a proof of the Collatz conjecture.