Four-digit numbers which are squared sums

Heather Coughlin and Brian Jue have written an article called Four-digit numbers which are squared sums. The article was recently published online in International Journal of Mathematical Education in Science and Technology. Here is the article abstract:

There is a very natural way to divide a four-digit number into 2 two-digit numbers. Applying an algorithm to this pair of numbers, determine how often the original four-digit number reappears.