It's an HP Touchsmart. They use laptop cpu's. This is a PC for a kitchen so it will not be a hard core gamer machine (while it will run some games) and run general wordprocessing apps etc along with Adobe Indesign and photoshop
Then get whatever's cheapest. If it's not for running heavy design software (like c4D) or for playing graphics heavy games, you won't need the power provided by either one of those, really.
Also, as far as I know, cores are just for multi-tasking. 4 cores can tackle more active programs and processes at one time than a dual core. But in terms of speed, a single core processor @ 3.0 GHz would probably run Crysis better than a quad core @ 2.0 GHz.
So yeah, if you won't be doing that, I'd go with whatever you can get for a song.