#### Vol. 12, No. 7, 2019

 Download this article For screen For printing  Recent Issues  The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Author Index Coming Soon Other MSP Journals  On generalizing happy numbers to fractional-base number systems

### Enrique Treviño and Mikita Zhylinski

Vol. 12 (2019), No. 7, 1143–1151
##### 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.

##### Keywords
happy numbers, fractional base, digital representation
Primary: 11A63