Fixed Points of Augmented Generalized Happy Functions

An augmented generalized happy function $S_{[c,b]}$ maps a positive integer to the sum of the squares of its base $b$ digits plus $c$. In this paper, we study various properties of the fixed points of $S_{[c,b]}$; count the number of fixed points of ${\mathrm{\S }}_{[c,b]}$, for $b\ge 2$ and $0<c<3b-3$; and prove that, for each $b\ge 2$, there exist arbitrarily many consecutive values of $c$ for which $S_{[c,b]}$ has no fixed point.