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

Abstract

The Collatz sequence for a given natural number $N$ is generated by repeatedly applying the map $N$ $\to$ $3N+1$ if $N$ is odd and $N$ $\to$ $N/2$ 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.

7 pages, 4 figures, 1 table; Table 1 has been fixed (an element was missing before the end of each sequence displayed); minor clarifications in the text