Smallest Cube whose Sum of Divisors is Cube

Theorem
The smallest cube $N$ such that $\sigma \left({N}\right)$ is also a cube is:
 * $27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209$

where $\sigma \left({N}\right)$ denotes the $\sigma$ function of $N$: the sum of the divisors of $N$

Proof
We have that:

Then from :