# Show that if $p$ is an odd prime, show a power $p^k$ can never be a perfect number

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?

• Can you describe the factors of $p^k$ precisely? – Sammy Black Oct 20 '14 at 18:12
• 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. – rogerl Oct 20 '14 at 18:13

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.