We will construct a sequence of positive integers $\{n_i\}_{(i \geqslant 1)}$ as follows $n_1 = 2$ and for all $r \geqslant 2$ we have $n_r = n_{r-1}$ if $3^r \mid n_{r-1}^3 + 10$ and one of the $3$ lifted solutions from $m^3 + 10 \equiv 0 \pmod {3^{r-1}}$ to $m^3 + 10 \equiv 1 \pmod {3^{r-1}}$ by Hensel's lemma. Now the contructed sequence satisifes $3^k \mid n_k^3 + 10$ and so choosing all $n_i$ with $i > k$, we are done.