Given a base integer
where is the least significant digit and is the most significant, if it is also the case that
then is a factorion. In other words, the sum of the factorials of the digits in a standard positional integer base (such as base 10) gives the same number as multiplying the digits by the appropriate power of that base. With the exception of 1, the factorial base representation of a factorion is always different from that in the integer base. Obviously, all numbers are factorions in factorial base.
1 is a factorion in any integer base. 2 is a factorion in all integer bases except binary. In base 10, there are only four factorions: 1, 2, 145 and 40585. For example, . (The factorial base representation of 40585 is 10021001).
- 1 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 125