Sequence of Dudeney Numbers

Theorem
The only positive integers which are equal to the sum of the digits of their cube are:
 * $0, 1, 8, 17, 18, 26, 27$

two of which are themselves cubes, and one of which is prime.

Proof
We have trivially that:

Then:

A quick empirical test shows that when $n = 46$, it is already too large to be the sum of the digits of its cube.

Also reported as
Some sources (either deliberately or by oversight) do not include $0$ in this list.

Also see

 * Definition:Armstrong Number, with which the numbers in this entry appear frequently to be conflated