Multiple of Perfect Number is Abundant

Theorem

Let $n$ be a perfect number.

Let $m$ be a positive integer such that $m > 1$.


Then $m n$ is abundant.


Proof

We have by definition of divisor sum function and perfect number that:

$\dfrac {\map {\sigma_1} n} n = 2$

But from Abundancy Index of Product is greater than Abundancy Index of Proper Factors:

$\dfrac {\map {\sigma_1} {m n} } {m n} > 2$

Hence the result by definition of abundant.

$\blacksquare$


Sources

  • 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $12$
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $12$
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