Smallest Cube whose Sum of Divisors is Cube

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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:

\(\ds N\) \(=\) \(\ds 27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209\)
\(\ds \) \(=\) \(\ds 3^9 \times 7^3 \times 11^3 \times 13^3 \times 17^3 \times 41^3 \times 43^3 \times 47^3 \times 443^3 \times 499^3 \times 3583^3\)
\(\ds \) \(=\) \(\ds 30 \, 154 \, 214 \, 043 \, 975 \, 990 \, 969^3\)


Then from $\sigma_1$ of $27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209$:

\(\ds \map {\sigma_1} N\) \(=\) \(\ds 65 \, 400 \, 948 \, 817 \, 364 \, 742 \, 403 \, 487 \, 616 \, 930 \, 512 \, 213 \, 536 \, 407 \, 552 \, 000 \, 000 \, 000 \, 000 \, 000\)
\(\ds \) \(=\) \(\ds 2^{39} \times 3^6 \times 5^{15} \times 7^3 \times 11^3 \times 13^3 \times 17^3 \times 29^3 \times 37^3 \times 61^3 \times 157^3\)
\(\ds \) \(=\) \(\ds 40 \, 289 \, 760 \, 243 \, 532 \, 800 \, 000^3\)


This theorem requires a proof.
In particular: It remains to be shown that this is the smallest such number.
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Historical Note

According to David Wells in his Curious and Interesting Numbers, 2nd ed. of $1997$, this result is due to (probably Frank) Rubin, and can be found in Journal of Recreational Mathematics, Volume $27$, on page $229$.

However, this has not been corroborated.


Sources

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