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 $\S_{ [c,b]}$, for $b \geq 2$ and $0<c<3b-3$; and prove that, for each $b \geq 2$, there exist arbitrarily many consecutive values of $c$ for which $S_{ [c,b]}$ has no fixed point.
Categories: math.NT