Consider the number $50$.
$50^2 = 2500 = 2^2 \times 5^4$, so $\phi(2500) = 2 \times 4 \times 5^3 = 8 \times 5^3 = 2^3 \times 5^3$. ¹
So $2500$ is a square and $\phi(2500)$ is a cube.

Find the sum of all numbers $n$, $1 \lt n \lt 10^{10}$ such that $\phi(n^2)$ is a cube.

¹ $\phi$ denotes Euler's totient function.