The question is still unclear to me, therefore I will outline what the options are.
The function $G(n)$ (well-defined for integers $n>1$, and positive for $n>3$) has the following properties:
Grönwall’s theorem: $$\limsup_{n\to\infty}G(n)=e^\gamma.$$ For a concrete infinite sequence $\{n_k\}$ for which $G(n_k)$ tends to $e^\gamma$, one may take $n_k=\mathrm{lcm}(1,\dots,k)$.
There are a bunch of numbers $n\le5040$ for which $G(n)>e^\gamma$.
Robin’s theorem: if If the Riemann hypothesis holds, then $G(n)< e^\gamma$ for all $n>5040$. If the Riemann hypothesis fails, there are infinitely many $n$ such that $G(n)>e^\gamma$ (and even $G(n)>e^\gamma+c/(\log n)^\beta$ for some constants $c,\beta>0$).
Another Robin’s theorem: $G(n)< e^\gamma+0.6482/(\log\log n)^2$ for all $n>1$ (unconditionally).
Therefore, there are the following possibilities:
Joro wants to find $n>1$ where $G(n)$ is maximal.
The answer is $\mathbf{n=3}$, which gives $G(3)\approx14.177183749182$. All other $G(n)$ are smaller by the unconditional Robin’s theorem above.
Joro wants to find $n>5040$ where $G(n)$ is maximal.
Case A: Riemann hypothesis holds.
The supremum of the sequence is $e^\gamma$, but all its elements are strictly smaller. Thus, there is no maximum.
Case B: Riemann hypothesis fails.
Let $n_0>5040$ be such that $G(n_0)>e^\gamma$. Then $G(n)< G(n_0)$ for all but finitely many $n$ by Grönwall’s theorem, hence the sequence does have a maximum, which is greater than $e^\gamma$. The point $n$ where it is achieved must be quite large, and cannot be exhibited explicitly at present, since this would amount to disproving the Riemann hypothesis. There may be more than one $n$ achieving the maximal $G(n)$, but in any case there are only finitely many such $n$.
Joro wants to exhibit an infinite sequence of $n$ on which $G(n)$ tends to $e^\gamma$.
(He already said he does not, but I think it’s actually a more natural question than the other two readings above.) There is no unique answer, one possibility is to take the numbers $n=\mathrm{lcm}(1,\dots,k)$ as above.
Joro wants something else,
in which case he should state the question more clearly.
