# 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 |
---|---|

Canonical name | Factorion |

Date of creation | 2013-03-22 17:43:52 |

Last modified on | 2013-03-22 17:43:52 |

Owner | CompositeFan (12809) |

Last modified by | CompositeFan (12809) |

Numerical id | 7 |

Author | CompositeFan (12809) |

Entry type | Definition |

Classification | msc 11A63 |

Classification | msc 05A10 |