Vol. 11, No. 2, 2018

Download this article
Download this article For screen
For printing
Recent Issues

Volume 13
Issue 3, 361–539
Issue 2, 181–360
Issue 1, 1–180

Volume 12, 8 issues

Volume 11, 5 issues

Volume 10, 5 issues

Volume 9, 5 issues

Volume 8, 5 issues

Volume 7, 6 issues

Volume 6, 4 issues

Volume 5, 4 issues

Volume 4, 4 issues

Volume 3, 4 issues

Volume 2, 5 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Scientific Advantages
Submission Guidelines
Submission Form
Ethics Statement
Editorial Login
ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
Author Index
Coming Soon
Other MSP Journals
This article is available for purchase or by subscription. See below.
Numbers and the heights of their happiness

May Mei and Andrew Read-McFarland

Vol. 11 (2018), No. 2, 235–241

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).

PDF Access Denied

However, your active subscription may be available on Project Euclid at

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 30.00:

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

Communicated by Kenneth S. Berenhaut
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