On generalizing happy numbers to fractional-base number systems

Treviño, Enrique; Zhylinski, Mikita

doi:10.2140/involve.2019.12.1143

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On generalizing happy numbers to fractional-base number systems

Enrique Treviño and Mikita Zhylinski

Vol. 12 (2019), No. 7, 1143–1151

DOI: 10.2140/involve.2019.12.1143

Abstract

Let $n$ be a positive integer and $S_{2} (n)$ 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., $S_{2}^{k} (n) = 1$ , 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 $S_{e, p ∕ q} (n)$ 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 $S_{e, p ∕ q}^{k} (n) = S_{e, p ∕ q}^{m} (n)$ . 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

Mathematical Subject Classification 2010

Primary: 11A63

Milestones

Received: 30 September 2018

Revised: 12 June 2019

Accepted: 22 June 2019

Published: 12 October 2019

Communicated by Kenneth S. Berenhaut

Authors

Enrique Treviño
Lake Forest College Lake Forest, IL United States

Mikita Zhylinski
Lake Forest College Lake Forest, IL United States

Vol. 12, No. 7, 2019