Divisor of Deficient Number is Deficient

Theorem

Let $n$ be a perfect number.

Let $n = k d$ where $r$ is a positive integer.


Then $k$ is deficient.


Proof

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

$\dfrac {\map {\sigma_1} {k d} } {k d} < 2$

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

$\dfrac {\map {\sigma_1} {k d} } {k d} > \dfrac {\map {\sigma_1} k} k$

That is:

$\dfrac {\map {\sigma_1} k} k < 2$

Hence the result by definition of deficient.

$\blacksquare$


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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