$n$-product-periodic topological spaces

We call an topological space $ (X,\tau)$ $ n$ product-periodic for an integer $ n\geq 3$ if $ \prod_{i=1}^n X \cong X$ but for all integers $ k$ with $ 2\leq k\leq n-1$ we have $ \prod_{i=1}^k X \not\cong X$ .

Is there an integer $ n\geq 3$ such that there is an $ n$ -product-periodic space, and is there an integer $ m\geq 3$ such that there is no $ m$ -product-periodic space?

$n$-product-periodic groups

We call an (infinite) group $ n$ product-periodic for an integer $ n\geq 3$ if $ \prod_{i=1}^n G \cong G$ but for all integers $ k$ with $ 2\leq k\leq n-1$ we have $ \prod_{i=1}^k G \not\cong G$ .

Is there an integer $ n\geq 3$ such that there is a group $ G$ such that $ G$ is $ n$ -product-periodic, and is there an integer $ m\geq 3$ such that there is no group $ G$ such that $ G$ is $ m$ -product-periodic?