I’ve been experimenting a little with home automation. I have one device that turns on with an RF remote, which just has a single button. On the back of the remote is printed 433/1527m. I think the 433 means MHz, but I’m not sure about the 1527m. Is that even something I need to know if, say, I want to buy an RF transmitter to duplicate what the remote is doing?
This is a bit of a leap, but the EV1527 chip is frequently paired with 433 MHz devices, acting as an encoder. I’m not sure, but snooping around it looks like the same protocol may be used by other chips (clones or not).
You can get pre-built units, like this one, using the chip. They can be paired with another device, so presumably you can use your existing remote to program the new one with the same codes.
Again, this is a bit of a leap based only on the 1527 number, but it seems reasonably likely. The 433 is definitely MHz, as you suspected. That’s a common unlicensed band for things like this.
Thanks, Doc! That definitely moves me one step forward. Your first link has 315Mhz in the URL, though interestingly not in the data sheet it links to. Why would the data sheet leave that out? Is it somehow irrelevant? It’s all still pretty baffling to me. I guess I could get that pre-built unit, pair it, then dismantle it and somehow integrate its guts into my circuit. Maybe it’s easier just to build a robot hand to push the button though. 
There are all kinds of setups in the RF world, but in this case there is a division of labor between the encoder chip and the radio chip. The EV1527 here can be tied to any of a few radios, likely including 315, 433, and 915 MHz (the only thing special about these frequencies is that they’re low frequency and open for use by low-power, non-continuous use–like remotes).
If you wanted to build something, the radio side would use something like one of these units. They take just a single bit input, basically on and off, and so need some kind of encoder (i.e., the EV1527) as the front-end if you want to do something more advanced. Or an Arduino, etc.