Higher associativity of Moore spectra

The Moore spectrum ${\mathrm{M}}_p(i)$ is the cofiber of the $p^i$ map on the sphere spectrum. For a fixed $p$ and $n$, we find a lower bound on $i$ for which ${\mathrm{M}}_p(i)$ is guaranteed to be $n$-fold associative. This bound depends on the stable homotopy groups of spheres.