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CYCLES AND FIXED POINTS OF HAPPY FUNCTIONS

By Kathryn Hargreaves and Samir Siksek

Abstract

Let N = {1, 2, 3, · · · } denote the natural numbers. Given integers e ≥ 1 and b ≥ 2, let x = ∑ n i=0 aib i with 0 ≤ ai ≤ b − 1 (thus ai are the digits of x in base b). We define the happy function Se,b: N − → N by Se,b(x) = A positive integer x is then said to be (e, b)-happy if Sr e,b (x) = 1 for some r ≥ 0, otherwise we say it is (e, b)-unhappy. In this paper we investigate the cycles and fixed points of the happy functions Se,b. We give an upper bound for the size of elements belonging to the cycles of Se,b. We also prove that the number of fixed points of S2,b i

Year: 2009
OAI identifier: oai:CiteSeerX.psu:10.1.1.173.5234
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