  A weakly consecutive sequence (WCS) is a permutation $\sigma$ of $\{1,
\ldots, k\}$ such that if an integer $d$ divides $\sigma(i)$, then $d$ also
divides $\sigma(i \pm d)$ insofar as these are defined. The structure of weakly
consecutive sequences is surprisingly rich, and it is difficult to find a
formula for the number $N(k)$ of WCS's of length $k$. However, for a given $k$
we describe four starting sequences, to each of which we can apply three
\emph{rules} or operations to generate new WCS's. We conjecture that any WCS
can be constructed by applying these rules, which depend in an intricate way on
the primality of $k$ and surrounding integers. We find bounds for $N(k)$ by
analyzing these rules.
