Abstract:

We generalize the concept of happy number as follows. Let ${\mathbf e}= (e_0,e_1,....)$ be a sequence with $e_0 = 2$ and $e_i = \{1,2\}$ for $i > 0$. Define $S_{{\mbox{\scriptsize $\mathbf e$}}}:{{\mathbb{Z}}}^+ \rightarrow {{\mathbb{Z}}}^+$ by

$$S_{{\mbox{\scriptsize$\mathbf e$}}}\left(\sum_{i=0}^n a_i 10^i \right) = \sum_{i=0}^n a_i^{e_i}.$$

If $S_{{\mbox{\scriptsize $\mathbf e$}}}^k(a) = 1$ for some $k \in {{\mathbb{Z}}}^+$, then we say that $a$ is a semihappy number or, more precisely, an ${\mathbf e}$-semihappy number. In this paper, we determine fixed points and cycles of the functions $S_{{\mbox{\scriptsize $\mathbf e$}}}$ and discuss heights of semihappy numbers. We also prove that for each choice of ${\mathbf e}$, there exist arbitrarily long finite sequences of consecutive ${\mathbf e}$-semihappy numbers.