This question is out of genuine curiosity. There is no reason to believe these numbers have any meaning. I am interested if there is any information related to the sequence, $$a_n=\sqrt[n]{n!}$$ I encountered the sequence when investigating values of $a$ such that the equation $$x!=a^x$$ has integer solutions. For example, $$x!=(a_5)^x=5!^{x/5}\Rightarrow x=5$$
More generally, where can I consult when I want to find information related to some set of numbers? I've seen oeis.org but from what I understand they only have information regards to integer sequences, which is not so helpful in my case.
Thanks in advance.
Using Stirling approximation $$a_n=\frac ne\left(1+ \frac{\log (2 \pi n)}{2 n}+\frac{3 \log ^2(2 \pi n)+2}{24 n^2}+\frac{\log (2 \pi n) \left(\log ^2(2 \pi n)+2\right)}{48 n^3}+O\left(\frac{1}{n^4}\right)\right)$$ is a very good approximation : the relative error is smaller than $1.0$% for $n>1$ and smaller than $0.1$% for $n>4$.