Fixed points of augmented generalized happy functions

Source 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.