Factorions Base 10

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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