It depends, as always, on your use case. If you're just browsing facebook or looking up pictures on instagram... Linux running on a Pentium M works fine. If you want to play The Latest Games at 4K 60fps™, you're going to need something a little bit more substantial.
For me, the vast majority of CPUs released in the past 15 years are powerful enough for me to do *something* productive with if I have nothing else to work with, but it *has* to have a solid-state drive. It doesn't have to be the fastest SSD out there, but anything I use that has something resembling a SATA connector will immediately get an SSD slapped into it, and if it doesn't...
I had a Motion Computing LE1700 with a Core Duo, and its drive interface was a tiny 40-pin IDE connector on a ribbon cable. So I got an IDE to mSATA adapter card. Super jank, but it was worth it.
The other (mostly philosophical) stipulation I have for systems I use is that they have to not be based on the Netburst architecture. Pentium 4s, Pentium Ds, Celeron Ds... the first Core 2 Duos for desktops ran circles around the Pentium Extreme Edition 965. Netburst was such a dumpster fire that Intel had to backpedal to the Pentium III for their future processors.
If I'm using a computer for general use and productivity, I prefer six cores or more.