This is a problem from my past Qual: "Prove or disapprove. There is an entire function $f$ s.t. $f(n)=0$ for all $n\in \mathbb{N}$ and nonzero elsewhere."
For this type of problems, I think of Identity Theorem. $f(n)=0$ for all $n$ and hence the set $\{z|f(z)=0 \}$ has an accumulation point ($\infty$), so $f=0$ and hence there is no such function.
However, I suppose $\infty$ does not count as an accumulation point? Am I right and if not, how should I solve this?
$\frac{1}{\Gamma(-z)}$ will do if you allow $n=0$ as a zero, a translation by $1$ of that if you start at $n=1$