Minimum current to prevent auto-shutdown

I want to use an battery to power a our product. But our product doesn’t consume enough current. So, the battery shuts down automatically after some time. Using a repeated pulse generator circuit, I can prevent this from happening. Could you please tell me the minimum current needed for Anker PowerCore mini 3350mAh and Astro E1 5200mAh? I can of course try to test with some battery samples with different current, pulse width and repetition frequency. But it would be much more easier, faster and safer if you could recommend these parameters.

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