Factorion 2008-01-08 18:25:13 2008-08-06 14:38:03 Definition factorial % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Given a base $b$ integer $$n = \sum_{i = 1}^k d_ib^{i - 1}$$ 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 \emph{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). \begin{thebibliography}{1} \bibitem{dw} D. Wells, {\it The Penguin Dictionary of Curious and Interesting Numbers} London: Penguin Group. (1987): 125 \end{thebibliography}