positive multiple of a semiperfect number is also semiperfect

Just as the theorem on multiples of abundant numbers shows that multiples 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 divisors $d_1,\mathrm{\dots },d_{k-1}$ of $n$ (where $k=\tau (n)$ and $\tau (x)$ is the divisor function), 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 $d_{1}m,\mathrm{\dots },d_{k-1}m$. With our smart iterator $i$ and thanks to the distributive property of multiplication, it follows that

$$\sum _{i=1}^{k-1}{d}_{i}m=nm,$$

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