Smallest Cube whose Sum of Divisors is Cube

Theorem
The smallest cube $N$ such that $\map {\sigma_1} N$ 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 $\map {\sigma_1} N$ denotes the divisor sum of $N$.

Proof
We have that:

Then from :