Vol. 12, No. 7, 2019

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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
Abstract

Let n be a positive integer and S2(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 S2 on n is 1, i.e., S2k(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 Se,pq(n) be the sum of the e-th powers of the digits of n base p q. Let k be the smallest nonnegative integer for which there exists a positive integer m > k satisfying Se,pqk(n) = Se,pqm(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