  An augmented generalized happy function $S_{[c,b]}$ maps a positive integer
to the sum of the squares of its base $b$ digits plus $c$. For $b\geq 2$ and $k
\in \mathbb{Z}^+$, a $k$-desert base $b$ is a set of $k$ consecutive
non-negative integers $c$ for each of which $S_{[c,b]}$ has no fixed points. In
this paper, we examine a complementary notion, a $k$-oasis base $b$, which we
define to be a set of $k$ consecutive non-negative integers $c$ for each of
which $S_{[c,b]}$ has a fixed point. In particular, after proving some basic
properties of oases base $b$, we compute bounds on the lengths of oases base
$b$ and compute the minimal examples of maximal length oases base $b$ for small
values of $b$.
