Semihappy Numbers

We generalize the concept of happy number as follows. Let ${\mathbf e}= (e_0,e_1,....)$ be a sequence with

and $e_i = \{1,2\}$ for

. Define $S_{{\mbox{\scriptsize $\mathbf e$}}}:{{\mathbb{Z}}}^+ \rightarrow {{\mathbb{Z}}}^+$ by

$\displaystyle S_{{\mbox{\scriptsize$\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

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.

Semihappy Numbers

H. G. Grundman Department of Mathematics Bryn Mawr College Bryn Mawr, PA 19010 USA

H. G. Grundman
Department of Mathematics
Bryn Mawr College
Bryn Mawr, PA 19010
USA