**Problem**

Find all the entire functions that satisfy $$n^2f(\dfrac{1}{n})^3+f(\dfrac{1}{n})=0 \space \space \forall \space n \in \mathbb N$$

My idea was to define a function $$g(z)=z^2f(z)^3+f(z)$$

Since $f$ is an entire function and $g$ is composition of entire functions, then $g$ is entire as well. The function $g$ vanishes on the sequence $\{\dfrac{1}{n}\}\_{n \in \mathbb N}$ and $\dfrac{1}{n} \to 0 \in \mathbb C$, then $g$ is identically zero (if a function $f$ defined on a region $\Omega$ vanishes on a sequence of different points with a limit point in $\Omega$, then $f$ is identically zero).

We have $$0=g=z^2f(z) ^3+f(z),$$ from here we obtain $$f(z)=-z^2f(z)^3$$

I have the feeling that I have not answered the complete solution, but I don't know what else I can say about $f$. I would appreciate some help to complete my solution or suggestions of another answer.