What is the theoretical upper bound of factorion numbers?

Question

Recently I read about factorion numbers. I understood that there are only 4 factorion numbers, but what is the theoretical range in which they can be? Is it $[0, +\infty]$ or a smaller upper range? And if it is smaller, why?

Answer 1

There's no one standard definition for 'theoretical range'; given that the numbers are all known, it's legitimate to say that their 'theoretical range' is $[1, 40585]$. I suspect what you're trying to ask is 'what sort of easy upper bound can we prove?' - and the Mathworld page gives a big hint there: the sum of the factorials of an $n$-digit number can't exceed $n\times9!$; since an $n$-digit number is at least as large as $10^{n-1}$, then there can't be any factorions of $n_0$ digits or more, where $n_0$ is the first number for which $10^{n-1} > n\times9!$ - since $9!=362880$, this comes down to $10^{n-1} > n\times 362880$, or about $n\times 3\times 10^5$. Trying $n=7$ here gives $10^6 > 6\times 362880 = 2177280$, which is false, but trying $n=8$ gives $10^7 > 7\times 362880 = 2540160$, which is correct - so $n_0$ = 8, and all factorions have at most 7 digits. From there it's straightforward to use brute-force search to find the exact set of solutions.