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You don't want to run an Nano on battery. The regulator, the LED and so on will burn to much energy.
What you want is a Pro Mini at 8MHz, so it can directly run on 3.3-5V from a battery, with no regulator in between.
After a quick scan, this seems to explain it quite well.
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You could use a step up converter, but that wastes a lot of power.
I strongly recommend this excellent tutorial on low power operation.
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