PracticalNumber 2004-02-27 06:14:53 2004-04-30 17:03:13 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 A positive integer $m$ is called a \emph{practical number} if every positive integer $n<m$ is a sum of distinct positive divisors of $m$. {\bf \PMlinkescapetext{Lemma}.} An integer $\,m\ge 2,$ $\,m=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_\ell^{\alpha_\ell},$ with primes $p_1<p_2<\dots<p_\ell$ and integers $\alpha_i\ge 1,$ is practical\index{practical} if and only if $\,p_1=2\,$ and, for $i=2,3,\dots,\ell,$ $$p_i\le\sigma\!\left(p_1^{\alpha_ 1}p_2^{\alpha_2}\cdots p_{i-1} ^{\alpha_{i-1}}\right)+1,$$ where $\sigma(n)$ denotes the sum of the positive divisors of $n.$ Let $P(x)$ be the counting function of practical numbers. Saias~\cite{Sai}, using suitable sieve methods introduced by Tenenbaum \cite{Te1,Te2}, proved a good estimate in terms of a Chebishev-type theorem: for suitable constants $c_1$ and $c_2$, $$c_1\frac x{\log x}<P(x)<c_2\frac x{\log x}.$$ In \cite{Me1} Melfi proved a Goldbach-type result showing that every even positive integer is a sum of two practical numbers, and that there exist infinitely many triplets of practical numbers of the form $m-2,m,m+2$. \begin{thebibliography}{9} \bibitem{Me1} G.~Melfi, On two conjectures about practical numbers, {\it J.~Number Theory} {\bf 56} (1996), 205--210. \bibitem{Sai} E.~Saias, Entiers \`a diviseurs denses 1, {\it J.~Number Theory} {\bf 62} (1997), 163--191. \bibitem{Te1} G.~Tenenbaum, Sur un probl\`eme de crible et ses applications, Ann. Sci. \'Ec. Norm. Sup. (4) {\bf 19 } (1986), 1--30. \bibitem{Te2} G.~Tenenbaum, Sur un probl\`eme de crible et ses applications, 2. Corrigendum et \'etude du graphe divisoriel, Ann. Sci. \'Ec. Norm. Sup. (4) {\bf 28 } (1995), 115--127. \end{thebibliography}