Numbers and the heights of their happiness

Mei, May; Read-McFarland, Andrew

doi:10.2140/involve.2018.11.235

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-4184 (online)

ISSN 1944-4176 (print)

Author index

To appear

Other MSP journals

Numbers and the heights of their happiness

May Mei and Andrew Read-McFarland

Vol. 11 (2018), No. 2, 235–241

DOI: 10.2140/involve.2018.11.235

Abstract

A generalized happy function, $S_{e, b}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>S</mi></mrow><mrow><mi>e</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msub></math>$ maps a positive integer to the sum of its base $b$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi></math>$ digits raised to the $e$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>e</mi></math>$ -th power. We say that $x$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>$ is a base- $b$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi></math>$ , $e$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>e</mi></math>$ -power, height- $h$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>h</mi></math>$ , $u$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>u</mi></math>$ -attracted number if $h$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>h</mi></math>$ is the smallest positive integer such that $S_{e, b}^{h} (x) = u$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mrow><mi>S</mi></mrow><mrow><mi>e</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow><mrow><mi>h</mi></mrow></msubsup><mrow><mo class="MathClass-open">(</mo><mrow><mi>x</mi></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-rel">=</mo> <mi>u</mi></math>$ . Happy numbers are then base-10, 2-power, 1-attracted numbers of any height. Let $σ_{h, e, b} (u)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>σ</mi></mrow><mrow><mi>h</mi><mo class="MathClass-punc">,</mo><mi>e</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>u</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ denote the smallest height- $h$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>h</mi></math>$ , $u$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>u</mi></math>$ -attracted number for a fixed base $b$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi></math>$ and exponent $e$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>e</mi></math>$ and let $g (e)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>g</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>e</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ denote the smallest number such that every integer can be written as $x_{1}^{e} + x_{2}^{e} + \dots + x_{g (e)}^{e}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>e</mi></mrow></msubsup> <mo class="MathClass-bin">+</mo> <msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>e</mi></mrow></msubsup> <mo class="MathClass-bin">+</mo> <mo class="MathClass-rel">⋯</mo> <mo class="MathClass-bin">+</mo> <msubsup><mrow><mi>x</mi></mrow><mrow><mi>g</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>e</mi></mrow><mo class="MathClass-close">)</mo></mrow></mrow><mrow><mi>e</mi></mrow></msubsup></math>$ for some nonnegative integers $x_{1}, x_{2}, \dots, x_{g (e)}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo class="MathClass-punc">,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo class="MathClass-punc">,</mo><mo class="MathClass-op">…</mo><mo class="MathClass-punc">,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>g</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>e</mi></mrow><mo class="MathClass-close">)</mo></mrow></mrow></msub></math>$ . We prove that if $p_{e, b}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>p</mi></mrow><mrow><mi>e</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msub></math>$ is the smallest nonnegative integer such that $b_{e, b}^{p_{e, b}} > g (e)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>b</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>e</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msub></mrow></msup> <mo class="MathClass-rel">&gt;</mo> <mi>g</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>e</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ ,

d=⎡⎢ ⎢ ⎢ ⎢⎢g(e)+11−(b−2b−1)e+e+pe,b⎤⎥ ⎥ ⎥ ⎥⎥,&lt;math xmlns="http://www.w3.org/1998/Math/MathML" display="block"&gt;&lt;mrow&gt; &lt;mi&gt;d&lt;/mi&gt; &lt;mo class="MathClass-rel"&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo class="MathClass-open" fence="true" mathsize="2.45em"&gt;⌈&lt;/mo&gt;&lt;mrow&gt; &lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mrow&gt;&lt;mo class="MathClass-open"&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mo class="MathClass-close"&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;mo class="MathClass-bin"&gt;+&lt;/mo&gt; &lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt; &lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt; &lt;mo class="MathClass-bin"&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo class="MathClass-open" fence="true" mathsize="1.19em"&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo class="MathClass-bin"&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt; &lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo class="MathClass-bin"&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;mo class="MathClass-close" fence="true" mathsize="1.19em"&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt; &lt;mo class="MathClass-bin"&gt;+&lt;/mo&gt; &lt;mi&gt;e&lt;/mi&gt; &lt;mo class="MathClass-bin"&gt;+&lt;/mo&gt; &lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo class="MathClass-punc"&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mspace width="0.3em" class="thinspace"/&gt;&lt;/mrow&gt;&lt;mo class="MathClass-close" fence="true" mathsize="2.45em"&gt;⌉&lt;/mo&gt;&lt;/mrow&gt;&lt;mo class="MathClass-punc"&gt;,&lt;/mo&gt; &lt;/mrow&gt;&lt;/math&gt;

and $σ_{h, e, b} (u) \geq b^{d}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>σ</mi></mrow><mrow><mi>h</mi><mo class="MathClass-punc">,</mo><mi>e</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>u</mi></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-rel">≥</mo> <msup><mrow><mi>b</mi></mrow><mrow><mi>d</mi></mrow></msup> </math>$ , then $S_{e, b} (σ_{h + 1, e, b} (u)) = σ_{h, e, b} (u)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>S</mi></mrow><mrow><mi>e</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>h</mi><mo class="MathClass-bin">+</mo><mn>1</mn><mo class="MathClass-punc">,</mo><mi>e</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>u</mi></mrow><mo class="MathClass-close">)</mo></mrow></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-rel">=</mo> <msub><mrow><mi>σ</mi></mrow><mrow><mi>h</mi><mo class="MathClass-punc">,</mo><mi>e</mi><mo class="MathClass-punc">,</mo><mi>b</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>u</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ .

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

Vol. 11, No. 2, 2018