The question is: Is there an entire function $f$ with $f(n)=n/(1+n^2)$ for each $n\in\mathbb N$?
This is how far I came: $g(z)=f(1/z)$ is holomorphic in $z\neq0$ with $g(1/n)=n/(1+n^2)$, so 0 cannot be removable by the Identity Theorem. If 0 is a pole, then $g(z)=h(z)/z^m$ for an entire function $h$ with $h(0)\neq0$ and some $m\in\mathbb N$. But then $h(1/n)=g(1/n)/n^m=1/(n^{m-1}+n^{m+1})\to0$ as $n\to\infty$ and so $h(0)=0$. Contradiction. Thus, 0 must be an essential singularity of $g$ (i.e. $f$ must be transcendental). I don't know how to proceed further. Any hints are highly appreciated. Thanks in advance!