SelfNumber 2006-05-31 16:24:54 2006-06-04 14:13:12 Definition % 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 An integer $n$ that in a given base $b$ lacks a digitaddition generator. Consider, for example, the integer 41 in base 10. It can be expressed as 34 + 3 + 4. For 42, however, there is no such digitaddition, hence it is a self number. If $2|b$, all odd $n < b$ will be self numbers. Though self numbers form a small proportion of most ranges of $2b$ consecutive integers, there are infinitely many of them: The recurrence relation $S_i = (b - 2)b^{i - 1} + S_{i - 1} + (b - 2)$ (with $S_1 = b - 1$ if $2|b$ and $S_1 = b - 2$ otherwise) will give an incomplete though infinite list of self numbers. Reference Kaprekar, D. R. {\em The Mathematics of New Self-Numbers}. Devaiali, 1963: 19 - 20