Multiple of Perfect Number is Abundant
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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): $12$