\documentclass[12pt]{article}

\usepackage{amsmath,amsfonts,amsthm}

\def\Z{{\mathbb Z}}
\newcommand{\e}{{\mathbf e}}
\newcommand{\se}{{\mbox{\scriptsize$\mathbf e$}}}


\begin{document}

\begin{abstract}
We generalize the concept of happy number as follows.
Let $\e = (e_0,e_1,....)$ be a sequence with $e_0 = 2$
and $e_i = \{1,2\}$ for $i > 0$.
Define $S_{\se}:{\Z}^+ \rightarrow {\Z}^+$ by
\[S_{\se}\left(\sum_{i=0}^n a_i 10^i \right) = \sum_{i=0}^n a_i^{e_i}.\]
If $S_{\se}^k(a) = 1$ for some $k \in {\Z}^+$, then we say that $a$ is
a {\em semihappy number} or, more precisely, an {\em $\e$-semihappy number}.
In this paper, we determine fixed points and cycles of the functions
$S_{\se}$ and discuss heights of semihappy numbers.
We also prove that for each choice of $\e$,
there exist arbitrarily long finite sequences of consecutive
$\e$-semihappy numbers.
\end{abstract}

\end{document}
