Positive Integers Equal to Sum of Digits of Cube

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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.

This sequence is A046459 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

We have trivially that:

\(\displaystyle 0^3\) \(=\) \(\displaystyle 0\)
\(\displaystyle 1^3\) \(=\) \(\displaystyle 1\)


Then:

\(\displaystyle 8^3\) \(=\) \(\displaystyle 512\)
\(\displaystyle 8\) \(=\) \(\displaystyle 5 + 1 + 2\)


\(\displaystyle 17^3\) \(=\) \(\displaystyle 4913\)
\(\displaystyle 17\) \(=\) \(\displaystyle 4 + 9 + 1 + 3\)


\(\displaystyle 18^3\) \(=\) \(\displaystyle 5832\)
\(\displaystyle 18\) \(=\) \(\displaystyle 5 + 8 + 3 + 2\)


\(\displaystyle 26^3\) \(=\) \(\displaystyle 17576\)
\(\displaystyle 26\) \(=\) \(\displaystyle 1 + 7 + 5 + 7 + 6\)


\(\displaystyle 27^3\) \(=\) \(\displaystyle 19683\)
\(\displaystyle 27\) \(=\) \(\displaystyle 1 + 9 + 6 + 8 + 3\)


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


Historical Note

David Wells report in his $1997$ work Curious and Interesting Numbers, 2nd ed. that this result can be found in an article by Monte James Zerger in Journal of Recreational Mathematics, volume $25$, page $248$, but this has not been corroborated, through want of evidence.

N.J.A. Sloane reports in A046459 of the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008) that the result originated with Claude Séraphin Moret-Blanc in $1879$.


Sources