Constructing an entire function $f(z)$ such that $f(0)=1$ and $f(n^{1/3}-n^{1/6})=0$ for every $n\geq 2$

Question

I would like to find an entire function satisfying $f(0)=1$ and $f(n^{1/3}-n^{1/6})=0$ for every $n\geq 2$. My initial thought was to use the Weierstrass Factorization Theorem which states if {$a_n$} is a sequence of distinct complex numbers such that $a_0=0$ and $a_n \rightarrow \infty$ as $n\rightarrow \infty$ and $k_n\in \mathbb{N}$ be such that $k_0 \geq 0$ and $k_n \geq 1$ for all other $n$, then there exists an entire function $f$ with zeroes at each $a_n$, nonzero elsewhere, and ord$(f;a_n)=k_n$ for each $n$. In particular, if {$p_n$} is any sequence of natural numbers such that $\displaystyle{\sum_{n=1}^\infty k_n\Big{(}\frac{r}{|a_n|}\Big{)}^{p_n+1}}$ converges for every $r>0$, then $z_0^{k_0}\displaystyle{\prod_{n=1}^\infty \Big{(}1-\frac{z}{a_n}\Big{)}^{k_n}exp\Big{(}k_n\sum_{j=1}^{p_n}\frac{1}{j}\Big{(}\frac{z}{a_j}\Big{)}^j \Big{)}}$ is the desired entire function. My confusion with this theorem comes with choosing the sequences of {$p_n$} and {$k_n$}. How do I find such sequences to apply with this theorem?