A Factorion is a number that equals the sum of the factorials of its decimal digits, e.g. 145 is a factorion because 1! + 4! + 5! = 1 + 24 + 120 = 145.
There are only 4 base 10 Factorions. A prior Perplexus problem, Factorials & factorions, asked you to find the fourth and largest one.

A "Factoritwo" is a base 10 integer such that the integer is equal to the sum of factorials of one greater than each digit.
For example, if we were checking 145 we calculate 2! + 5! + 6! which of course is not 145.

What is the smallest Factoritwo?