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3
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'factorion'
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| Title of object: |
factorion |
| Canonical Name: |
Factorion |
| Type: |
Definition |
| Created on: |
2008-01-08 18:25:13 |
| Modified on: |
2008-01-11 17:26:34 |
| Classification: |
msc:05A10, msc:11B65, msc:11A63 |
Revision comment (for changes between this and next version):
| MSC 11B65 removed (no change to content) |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
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%\usepackage{xypic}
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Content:
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} |
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