positive multiple of a semiperfect number is also semiperfect

Just as the theorem on multiples of abundant numbers shows that multiplesMathworldPlanetmath of abundant numbers are also abundant, it is also true that multiples of semiperfect numbers are also semiperfect, and T. Foregger’s proof of the abundant number theorem lays bare a simple mechanism that we can also employ for semiperfect numbers.

Given the divisorsMathworldPlanetmath d1,,dk-1 of n (where k=τ(n) and τ(x) is the divisor functionMathworldPlanetmath), sorted in ascending order for our convenience, and with a smart iterator i that somehow knows to skip over those divisors that contribute to n’s abundance, we can show that the divisors of nm (with m>0) will include d1m,,dk-1m. With our smart iterator i and thanks to the distributive property of multiplication, it follows that


our desired result.

Title positive multiple of a semiperfect number is also semiperfect
Canonical name PositiveMultipleOfASemiperfectNumberIsAlsoSemiperfect
Date of creation 2013-03-22 16:18:57
Last modified on 2013-03-22 16:18:57
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 4
Author PrimeFan (13766)
Entry type Derivation
Classification msc 11A05