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Show that if $p$ is an odd prime, show a power $p^k$ can never be a perfect number. I am little confused about this problem, any insight?

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    $\begingroup$ Can you describe the factors of $p^k$ precisely? $\endgroup$ – Sammy Black Oct 20 '14 at 18:12
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    $\begingroup$ And once you've done that, can you add them up? If you do so and stare at the result, you should see the answer. $\endgroup$ – rogerl Oct 20 '14 at 18:13
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Denote the sum of dividors of $p^k$ with $\sigma(p^k)$.Then $$\sigma(p^k)=p^k+p^{k-1}+p^{k-2}+...+p+1=p^k+\frac{p^k-1}{p-1}<p^k+p^k=2p^k$$
So $p^k$ is not perfect.

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