The 3n+1 problem: a partition of interest

A mapping conjugate to the Collatz mapping seems to imply that $\N=\{1,2,3,\ldots\}$ is partitioned in a trivial loop $\{1\}$ and `strings' that are ordered subsets of $\{\N \setminus 1\}$ that run from an element of $\{2+3\0\}$ to an element of $\{3+4\0\}$ ($\0=0 \cup \N$). In particular, this means that all trajectories except for the trivial loop go through an element of $\{3+4\0\}$ ($\{5+8\0\}$ for the original mapping). I give reasons for this conjecture. Next, I note that the 3n+1 numbers and the 3n+3 numbers are the only numbers from the generalization $3n+p, p \in \{\ldots,-3,-1,1,3,\ldots\}$ for which such a partition seems to exist. Suspiciously, these are also the only members for which the conjecture (reduction to the trivial loop) seems to hold.