Number which is Sum of Subfactorials of Digits
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Theorem
The only integer which is the sum of the subfactorials of its digits is $148 \, 349$:
- $148 \, 349 = \mathop !1 + \mathop !4 + \mathop !8 + \mathop !3 \mathop + \mathop !4 \mathop + \mathop !9$
Proof
We have:
\(\ds 148 \, 349\) | \(=\) | \(\ds 0 + 9 + 14 \, 833 + 2 + 9 + 133 \, 496\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \mathop !1 + \mathop !4 + \mathop !8 + \mathop !3 \mathop + \mathop !4 \mathop + \mathop !9\) |
A computer search can verify solutions under $10^6$ (that is, with no more than $6$ digits) in seconds.
Let $n$ be a $k$-digit number, for $k \ge 7$.
Then the sum of the subfactorials of its digits is not more than $\mathop !9 \times k$.
But we have:
\(\ds n\) | \(\ge\) | \(\ds 10^{k - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 10^6 \times 10^{k - 7}\) | ||||||||||||
\(\ds \) | \(\ge\) | \(\ds 10^6 \times \paren {1 + 9 \paren {k - 7} }\) | Bernoulli's Inequality | |||||||||||
\(\ds \) | \(>\) | \(\ds 7 \times \mathop !9 \times \paren {9 k - 62}\) | $7 \times \mathop !9 = 934472$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \mathop !9 \paren {63 k - 62 \times 7}\) | ||||||||||||
\(\ds \) | \(\ge\) | \(\ds \mathop !9 \times k\) | $k \ge 7$ | |||||||||||
\(\ds \) | \(\ge\) | \(\ds \text{sum of the subfactorials of digits of } n\) |
So no more numbers have this property.
$\blacksquare$
Historical Note
This result was apparently obtained by R.S. Dougherty, but details are hard to come by.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $148,349$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $148,349$