# factorion

Given a base $b$ integer

 $n=\sum_{i=1}^{k}d_{i}b^{i-1}$

where $d_{1}$ is the least significant digit and $d_{k}$ is the most significant, if it is also the case that

 $n=\sum_{i=1}^{k}d_{i}!$

then $n$ is a factorion. In other words, the sum of the factorials of the digits in a standard positional integer base $b$ (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, $4\times 10^{4}+0\times 10^{3}+5\times 10^{2}+8\times 10^{1}+5\times 10^{0}=4!+% 0!+5!+8!+5!=40585$. (The factorial base representation of 40585 is 10021001).

## References

• 1 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 125
Title factorion Factorion 2013-03-22 17:43:52 2013-03-22 17:43:52 CompositeFan (12809) CompositeFan (12809) 7 CompositeFan (12809) Definition msc 11A63 msc 05A10