# Positive Integers Equal to Sum of Digits of Cube

## 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

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

## 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

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $17$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $18$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $26$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $27$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $4913$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $17$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $18$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $26$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $27$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $4913$

- Weisstein, Eric W. "Cubic Number." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/CubicNumber.html