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

 Download this article For screen For printing  Recent Issues  The Journal About the Journal Editorial Board Subscriptions Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Ethics Statement Editorial Login 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