Yet another name for regular numbers / Hamming numbers / 5-smooth numbers. I don't think they're ugly, but arun apparently does.
I think they are rather nice. I'd leave a message for him on his blog to give him the "algorithm" that I use, but who wants to create a new login on every blog just to respond. Instead, I'll just leave it here.
It's at the bottom of this recent post of mine having to do with sexagesimal numbers.
Thanks for the link! My own code for the problem is on this earlier post. See also the links livejournal user leonardo_m left there; it's quite a famous and well-studied problem in functional programming circles. Since your post is partly about multiplying and dividing sexagesimals, you might also like this post if you haven't already seen it.