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 and a non-negative integer~$c$. A positive integer $u$ is in a \textit {cycle} of $S_{[c,b]}$ if, for some positive integer~$k$, ${S_{[c,b]}}^k(u)=u$, and, for positive integers $v$ and $w$, $v$ is $w$-\textit {attracted} for $S_{[c,b]}$ if, for some non-negative integer~$\ell$, ${S_{[c,b]}}^{\ell }(v)=w$. In this paper, we prove that, for each $c\ge 0$ and $b\ge 2$, and for any $u$ in a cycle of $S_{[c,b]}$: (1)~if $b$ is even, then there exist arbitrarily long sequences of consecutive $u$-attracted integers, and (2)~if $b$ is odd, then there exist arbitrarily long sequences of 2-consecutive $u$-attracted integers.