You can use Linux for this sort of thing, but you have to be very careful with the software stack. This is identical to a modern laptop where you can hit, say,

You can use Linux for this sort of thing, but you have to be very careful with the software stack. This is identical to a modern laptop where you can hit, say, 10 hours battery life, but the software must be asleep practically all of the time. Anything misbehaving (e.g. Flash, mds) will keep the CPU switched on and waste power. Smaller micros are less efficient per unit of work but they just aren't capable of running as fast and chewing up a lot of power. You can also compensate for sloppy code by turning down the clock rate. If you're running a BLE device (not Bluetooth Classic), 500mAh for a week of runtime sounds doable. An iPhone battery is 2-3Ah, and so long as cell modem and screen are switched off, I believe that it would hit a week of runtime.