**superperfect** numbers is in the form $2^k$ when $2^{k+1}-1$ is a Mersenne prime.I can prove that numbers in this form will be **superperfect** but cannot prove the converse that all **superperfect** numbers are in this form. can anybody help me to prove? If this is not true then also tell me I am not sure about it.

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## Comments

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TopNewestYou mean you saw it on wiki? 'Cause its there...

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I read it on Elementary Number Theory by David Burton,then opened wiki and saw but there is no proof there neither direct one nor converse

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This might be helpful...check out perfect Numbers, as explained by Numberphile: (http://m.youtube.com/watch? v=q8n15q1v4Xo)

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Thank you very much it became really useful to me .Though here they only proved the form of perfect number but listening to the class I got idea to the form of superperfect number.Identified the series of super perfect number

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Can you please provide a proof that if $2^{k+1} -1$ is a Mersenne prime then $2^{k}$ is a perfect number because that might help us when we try to prove the converse.

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Here let $n=2^k$

$\sigma(\sigma(2^k)$

=$\sigma(\frac {2^{k+1}-1}{2-1})$

=$\sigma(2^{k+1}-1)$

As $2^{k+1}-1$ is a prime $\sigma(2^{k+1}-1)=(2^{k+1}-1)+1=2^{k+1}=2.2^k=2n$

Proved that it is always a superperfect number.

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Certainly I will post it .just wait

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