TITLE: What is the importance of an integer sequence like the happy numbers?
QUESTION [6 upvotes]: I've been looking at the happy numbers which led me to the OEIS and showed me that there are many documented integer sequences. What I don't understand is the importance of these sequences. To me, the happy number just look like an interesting algorithm that you can apply to a number and decide something about it. Does the sequence of happy numbers have a deeper meaning? 
Thanks!

REPLY [5 votes]: The question has been thoroughly answered, I just want to show you why I think the happy number sequence is cool. Here is a graph I generated of all $n<100,000$ connected to the sum of its digits squared. Really beautiful.