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 $\S_{[c,b]}$, 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. 
Breeanne Baker Swart, Kristen A. Beck, Susan Crook, Christina   Eubanks-Turner, Helen G. Grundman, May Mei, Laurie Zack
math.NT