Factorions Base 10
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:
\(\ds 1\) | \(=\) | \(\ds 1!\) | ||||||||||||
\(\ds 2\) | \(=\) | \(\ds 2!\) | ||||||||||||
\(\ds 145\) | \(=\) | \(\ds 1 + 24 + 120\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1! + 4! + 5!\) | ||||||||||||
\(\ds 40 \, 585\) | \(=\) | \(\ds 24 + 1 + 120 + 40 \, 320 + 120\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4! + 0! + 5! + 8! + 5!\) |
A computer search can verify solutions under $2540160 = 9! \times 7$ in seconds.
Let $n$ be a $7$-digit number with $n > 2540160$.
Then the sum of the factorials of its digits is not more than $9! \times 7 = 2540160$.
So $n$ cannot be a factorion base $10$.
Now let $n$ be a $k$-digit number, for $k \ge 8$.
Then the sum of the factorials of its digits is not more than $9! \times k$.
But we have:
\(\ds n\) | \(\ge\) | \(\ds 10^{k - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 10^7 \times 10^{k - 8}\) | ||||||||||||
\(\ds \) | \(\ge\) | \(\ds 10^7 \times \paren {1 + 9 \paren {k - 8} }\) | Bernoulli's Inequality | |||||||||||
\(\ds \) | \(>\) | \(\ds 8 \times 9! \times \paren {9 k - 71}\) | $8 \times 9! = 2903040$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 9! \paren {72 k - 71 \times 8}\) | ||||||||||||
\(\ds \) | \(>\) | \(\ds 9! \times k\) | $k \ge 8$ | |||||||||||
\(\ds \) | \(\ge\) | \(\ds \text{sum of the factorials of digits of } n\) |
So there are no more factorions base $10$.
$\blacksquare$
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): $145$