Number which is Sum of Subfactorials of Digits

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.