Editing Happy number (section)

{{short description|Numbers with a certain property involving recursive summation}}
{{distinguish|text=[[Harshad number]] (derived from Sanskrit ''harśa'' meaning "great joy")}}
{{Use dmy dates|date=October 2020}}
[[File:DessinArbreHeureux01.png|thumb|Tree showing all happy numbers up to 100, with 130 seen with 13 and 31]]
In [[number theory]], a '''happy number''' is a number which eventually reaches 1 when the number is replaced by the sum of the square of each digit. For instance, 13 is a happy number because <math>1^2+3^2=10</math>, and <math>1^2+0^2=1</math>. On the other hand, 4 is not a happy number because the sequence starting with <math>4^2=16</math> and <math>1^2+6^2=37</math> eventually reaches <math>2^2+0^2=4</math>, the number that started the sequence, and so the process continues in an infinite cycle without ever reaching 1. A number which is not happy is called '''sad''' or '''unhappy'''.

More generally, a <math>b</math>-'''happy number''' is a [[natural number]] in a given [[number base]] <math>b</math> that eventually reaches 1 when iterated over the [[Perfect digital invariant|perfect digital invariant function]] for <math>p = 2</math>.<ref>{{cite web|url=http://mathworld.wolfram.com/SadNumber.html|title=Sad Number|publisher=Wolfram Research, Inc.|access-date=2009-09-16}}</ref>

The origin of happy numbers is not clear.  Happy numbers were brought to the attention of [[Reg Allenby]] (a British author and senior lecturer in [[pure mathematics]] at [[Leeds University]]) by his daughter, who had learned of them at school.  However, they "may have originated in Russia" {{harvcol|Guy|2004|p=§E34}}.