One of my favourite investigations. It is accessible for a range of abilities and offers a great lesson in the value of recording results carefully so you can use previous results and save yourself work!
An attractive statement of the problem on Transum
On TES Resources, Owen134866 has created a very clear PowerPoint to introduce the investigation.
Try this Happy Number Checker on Scratch by Eulguass. Select the green flag to use the program.
Douglas Twitchell – Articles for Educators – Mathematical Fun With Happy Numbers
Happy Numbers on Kids, Code and Computer Science. Some very useful observations here.
…and a little more advanced on Wolfram MathWorld!
Some rather more advanced reading from Cornell University Library!
On the Density of Happy Numbers by Justin Gilmer
And from Numberphile
Finally – from Dr Who we have Happy Primes!