practical number


The second definition of a practical numberPlanetmathPlanetmath is: a positive integer n such that each smaller integer m can be represented as a sum of distinct proper divisors di of n (for 1<i<τ(n), with τ(n) being the divisor functionDlmfDlmfMathworldPlanetmath; 1 is not considered a proper divisor for this application). The first few are 6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 72, 78, 80, 84, 88, 90, 96, 100, etc., listed in A007620 of Sloane’s OEIS.

For example, 12 is practical. It has for divisorsMathworldPlanetmathPlanetmath 1, 2, 3, 4, 6 and 12, but only 2, 3, 4 and 6 are considered proper divisors here. The sum can consist of a single summand, so we need only concern ourselves with numbers less than 12 that are not divisors of 12. We verify that indeed 2 + 3 = 5, 3 + 4 = 7, 2 + 6 = 8, 3 + 6 = 9, 4 + 6 = 10 and 2 + 3 + 6 = 11.

Under this definition, the powers of 2 are not practical numbers. Representing odd numbersMathworldPlanetmathPlanetmath smaller than a power of 2 requires using 1 in the sums of divisors.

Title practical number
Canonical name PracticalNumber1
Date of creation 2013-03-22 18:07:03
Last modified on 2013-03-22 18:07:03
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 5
Author CompositeFan (12809)
Entry type Definition
Classification msc 11A25
Related topic ImpracticalNumber