On generalizing happy numbers to fractional-base number systems

Abstract

Let $n$ be a positive integer and $S_2\left(n\right)$ be the sum of the squares of its decimal digits. When there exists a positive integer $k$ such that the $k$-th iterate of $S_2$ on $n$ is 1, i.e., $S_2^k\left(n\right)=1$, then $n$ is called a happy number. The notion of happy numbers has been generalized to different bases, different powers and even negative bases. In this article we consider generalizations to fractional number bases. Let $S_{e,p∕q}\left(n\right)$ be the sum of the $e$-th powers of the digits of $n$ base $\frac{p}{q}$. Let $k$ be the smallest nonnegative integer for which there exists a positive integer $m>k$ satisfying $S_{e,p∕q}^k\left(n\right)=S_{e,p∕q}^m\left(n\right)$. We prove that such a $k$, called the height of $n$, exists for all $n$, and that, if $q=2$ or $e=1$, then $k$ can be arbitrarily large.