Or in other words for which minimal n the next inequality always follows for any a>0:
just pick .
Well first thing first, what I wrote is the lowest n possible (there isn't a maximum cause you can always pick another one greater by adding 1 to the preceding number).
I am unaware of a closed analytical formula for what I wrote, I guess only Stirling approximation can work here (if you want to find some closed formula though still an approximation).