practical number

A positive integer m is called a practical numberPlanetmathPlanetmath if every positive integer n<m is a sum of distinct positive divisorsMathworldPlanetmathPlanetmath of m.

. An integer m2, m=p1α1p2α2pα, with primes p1<p2<<p and integers αi1, is practical if and only if p1=2 and, for i=2,3,,,


where σ(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 c1 and c2,


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.


  • 1 G. Melfi, On two conjectures about practical numbers, J. Number TheoryMathworldPlanetmathPlanetmath 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