Largest divisible subgroup not an intersection

Is there an abelian group $$G$$ for which the largest divisible subgroup of $$G$$ (given by the sum of all divisible subgroups) is not the intersection of all subgroups of the form $$nG$$ over the positive integers $$n$$?

Note that if $$ny=x \in G$$, then for any positive integer $$m$$, if $$z \in G$$ and $$nmz=x$$, then of course $$nmz=ny$$, but one cannot in general "cancel" the $$n$$ to get $$mz=y$$. Cancellation is possible only if one knows that $$0$$ is the only element of $$G$$ whose order divides $$n$$, which in particular holds if $$G$$ is torsionfree.

Let $$F = \bigoplus_{\Bbb N} \Bbb Z$$. Define $$H \leqslant F$$ by $$\langle e_1 - 2e_2, e_1 - 3e_3, \ldots \rangle$$.
Set $$G = F/H$$.
Then, $$\overline{e_1} \in nG$$ for all $$n \geqslant 1$$.
On the other hand, given any $$x \in F$$ with $$\overline{e_1} = 2 \overline{x}$$, there exists $$n \in \Bbb N$$ such that $$x \notin nG$$.
This shows that $$\overline{e_1}$$ cannot be in any divisible subgroup of $$G$$, even though it is in the intersection of all $$nG$$.