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I'm looking for a positive real $n$, given positive real $m>2$. The calculations for $n$ proceed according to the following diagram:

$$ \begin{array}{ccc} m & \rightarrow & 2^m \\ & & \downarrow \\ n & \leftarrow & p_n \# \end{array} $$

To explain, we start with $m$, calculate $2^m$ and solve for $n$ by setting $2^m = p_n\#$. Here, $p_n\#$ is the $n$th primorial, calculated by multiplying the first $n$ primes together. For example, $p_4\# = 2 \cdot 3 \cdot 5 \cdot 7$.

According to Wikipedia's entry on primorials, we have the following bound:

$${p_n}\# = {e^{(1 + o(1))n\ln (n)}}$$

I'm wondering if someone can find a way to estimate $n$, given $m$. Also, I'm hoping that someone can talk me through the steps, because I plan on doing a similar calculation with the primorials.

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