# practical number

A positive integer $m$ is called a practical number if every positive integer $n 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} 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}

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, 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 PracticalNumber 2013-03-22 14:12:17 2013-03-22 14:12:17 mathcam (2727) mathcam (2727) 6 mathcam (2727) Definition msc 11A25