Variations on a Theme of Collatz

Consider the recursive relation generating a new positive integer $n_{\mathrm{\ell }+1}$ from the positive integer $n_{\mathrm{\ell }}$ according to the following simple rules: if the integer $n_{\mathrm{\ell }}$ is odd, $n_{\mathrm{\ell }+1}=3n_{\mathrm{\ell }}+1$; if the integer $n_{\mathrm{\ell }}$ is even, $n_{\mathrm{\ell }+1}=n_{\mathrm{\ell }}\mathrm{/}2$. The so-called Collatz conjecture states that, starting from any positive integer $N$, the recursion characterized by the continued application of these rules ends up in the cycle $4,$ $2,1$. This conjecture is generally believed to be true (on the basis of extensive numerical checks), but it is as yet unproven. In this paper – based on the assumption that the Collatz conjecture is indeed true – we present a quite simple extension of it, which entails the possibility to divide all natural numbers into $3$ disjoint classes, to each of which we conjecture – on the basis of (not very extensive) numerical checks – $1\mathrm{/}3$ of all natural numbers belong; or, somewhat equivalently, to $2$ disjoint classes, to which we conjecture that respectively $1\mathrm{/}3$ and $2\mathrm{/}3$ of all natural numbers belong.