# Factorions Base 10

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## Contents

## Theorem

The following positive integers are the only factorions base $10$:

- $1, 2, 145, 40 \, 585$

This sequence is A014080 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Proof

From examples of factorials:

\(\displaystyle 1\) | \(=\) | \(\displaystyle 1!\) | |||||||||||

\(\displaystyle 2\) | \(=\) | \(\displaystyle 2!\) | |||||||||||

\(\displaystyle 145\) | \(=\) | \(\displaystyle 1 + 24 + 120\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1! + 4! + 5!\) | |||||||||||

\(\displaystyle 40 \, 585\) | \(=\) | \(\displaystyle 24 + 1 + 120 + 40 \, 320 + 120\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 4! + 0! + 5! + 8! + 5!\) |

## Historical Note

The fact that $40 \, 585$ is a factorion base $10$ was discovered as late as $1964$ by Leigh Janes.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $145$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $145$