Fixed points of augmented generalized happy functions

Abstract

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 pro\-perties of the fixed points of $S_{[c,b]} $; count the number of fixed points of $S_{[c,b]} $ for $b \geq 2$ and $0\lt c\lt 3b-3$; 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.