practical number

A positive integer $m$ is called a practical number if every positive integer $n<m$ is a sum of distinct positive divisors of $m$ .

. An integer $\,m\geq 2,$ $\,m=p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\cdots p_{\ell}^{\alpha_{\ell}},$ with primes $p_{1}<p_{2}<\dots<p_{\ell}$ and integers $\alpha_{i}\geq 1,$ is practical if and only if $\,p_{1}=2\,$ and, for $i=2,3,\dots,\ell,$

p_{i}\leq\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 [2], using suitable sieve methods introduced by Tenenbaum [3, 4], 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 [1] 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 $m-2,m,m+2$ .

References

1 G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996), 205–210.
2 E. Saias, Entiers à diviseurs denses 1, J. Number Theory 62 (1997), 163–191.
3 G. Tenenbaum, Sur un problème de crible et ses applications, Ann. Sci. Éc. Norm. Sup. (4) 19 (1986), 1–30.
4 G. Tenenbaum, Sur un problème de crible et ses applications, 2. Corrigendum et étude du graphe divisoriel, Ann. Sci. Éc. Norm. Sup. (4) 28 (1995), 115–127.

Title	practical number
Canonical name	PracticalNumber
Date of creation	2013-03-22 14:12:17
Last modified on	2013-03-22 14:12:17
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	6
Author	mathcam (2727)
Entry type	Definition
Classification	msc 11A25