  The $3x+k$ function $T_{k}(n)$ sends $n$ to $(3n+k)/2$ resp. $n/2,$ according
as $n$ is odd, resp. even, where $k \equiv \pm 1~(\bmod \, 6)$. The map
$T_k(\cdot)$ sends integers to integers, and for $m \ge 1$ let $n \rightarrow
m$ mean that $m$ is in the forward orbit of $n$ under iteration of
$T_k(\cdot).$ We consider the generating functions $f_{k,m}(z) = \sum_{n>0, n
\rightarrow m} z^{n},$ which are holomorphic in the unit disk. We give
sufficient conditions on $(k,m)$ for the functions $f_{k, m}(z)$ have the unit
circle $\{|z|=1\}$ as a natural boundary to analytic continuation. For the
$3x+1$ function these conditions hold for all $m \ge 1$ to show that
$f_{1,m}(z)$ has the unit circle as a natural boundary except possibly for $m=
1, 2, 4$ and $8$. The $3x+1$ Conjecture is equivalent to the assertion that
$f_{1, m}(z)$ is a rational function of $z$ for the remaining values $m=1,2, 4,
8$.
