Vol. 11, No. 2, 2018

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Numbers and the heights of their happiness

May Mei and Andrew Read-McFarland

Vol. 11 (2018), No. 2, 235–241
Abstract

A generalized happy function, Se,b maps a positive integer to the sum of its base b digits raised to the e-th power. We say that x is a base-b, e-power, height-h, u-attracted number if h is the smallest positive integer such that Se,bh(x) = u. Happy numbers are then base-10, 2-power, 1-attracted numbers of any height. Let σh,e,b(u) denote the smallest height-h, u-attracted number for a fixed base b and exponent e and let g(e) denote the smallest number such that every integer can be written as x1e + x2e + + xg(e)e for some nonnegative integers x1,x2,,xg(e). We prove that if pe,b is the smallest nonnegative integer such that bpe,b > g(e),

d = g(e) + 1 1 (b2 b1)e + e + pe,b,

and σh,e,b(u) bd , then Se,b(σh+1,e,b(u)) = σh,e,b(u).

Keywords
happy numbers, integer sequences, iteration, integer functions
Mathematical Subject Classification 2010
Primary: 11A99, 11A63
Milestones
Received: 4 November 2015
Revised: 5 April 2017
Accepted: 9 May 2017
Published: 17 September 2017

Communicated by Kenneth S. Berenhaut
Authors
May Mei
Department of Mathematics & Computer Science
Denison University
Granville, OH
United States
Andrew Read-McFarland
Department of Mathematics & Computer Science
Denison University
Granville, OH
United States