For any natural number $n$, we have $n^{2} \equiv 0$ or $n^{2} \equiv 1(\bmod 4)$ and $n^{3} \equiv 0$ or $n^{3} \equiv \pm 1(\bmod 9)$. Thus $\{36 n+3 \mid n=1,2, \ldots\}$ is a progression with the required property.
Find an infinite non-constant arithmetic progression of positive integers such that each term is neither a sum of two squares, nor a sum of two cubes (of positive integers).