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).