On generalizing happy numbers to fractional-base number systems

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,p∕q(n) be the sum of the e-th powers of the digits of n base pq. Let k be the smallest nonnegative integer for which there exists a positive integer m>k satisfying Se,p∕qk(n)=Se,p∕qm(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.