Is there a lower bound for the nth primorial?

Question

Can $n\#$ (product of all primes less or equal to n) be bounded from below by some lower bound such as $2^n$?

Answer 1

OEIS A002110 (it would be more appropriate if it were A002310) lists the primorials but indexes them in order, not by the highest prime. It says that the values are $\left( m^m\right)^{1+o(1)}$. We can use the fact that the prime counting function is a little greater than $\frac n{\log n}$ to say that $n\# \gt \left(\frac n{\log n}\right)^{\frac n{\log n}}$ for $n$ prime