Henry Ernest Dudeney/Modern Puzzles/69 - Root Extraction/Solution

Modern Puzzles by Henry Ernest Dudeney: $69$

In a conversation I had with Professor Simon Greathead, the eminent mathematician, ...

the extraction of the cube root.

"Ah," said the professor, "it is astounding what ignorance prevails ...

... the simple fact that, to extract the cube root of a number, all you have to do is to add together the digits.

Thus, ignoring the obvious case of the number $1$, if we want the cube root of $512$, add the digits -- $8$, and there you are!"

I suggested that that was a special case.

"Not at all," he replied. "Take another number at random -- $4913$ -- and the digits add to $17$, the cube of which is $4913$."

I did not presume to argue the point with the learned man,

but I will just ask the reader to discover all the other numbers whose cube root is the same as the sum of their digits.

The other numbers are:

$\ds 5832$

$=$

$\ds 18^3$

$\ds 17 \, 576$

$=$

$\ds 26^3$

$\ds 19 \, 683$

$=$

$\ds 27^3$