In many ways it depends on how you shoot, and really only you can answer this.
I was content with my crop and a 24-60mm f/2.8 (Sigma), but also had a 10-20mm (also Sigma). (But) To this point I do not typically shoot wide, so for me 24mm was often wide enough. I was able to deal with the missing range between 20 to 24. That range is not nearly as big as 16 to 24mm that you (the OP) would have.
(It also may be pertinent) I was also shooting film at the time so wanted a lens that I could use on both bodies.
For many people 24mm (on a crop) will not be wide enough. I do own a good 24-105. While I have not used a 15-85, I've seen images from it. From what I've seen, I believe these two lens are very close in IQ used on a crop body. If I only were shooting a crop camera body, I'd rather go with the 15-85 over the 24-105. (In reality, I'd opt for a f/2.8 lens though.)
My opinion is to get glass for the body you currently own, tempered by how you shoot. IMO of the two, the 15-85 is a better lens for most people a crop body. (YMMV)