Are you Perfect or Superperfect?

Number Theory Level 2

How many Perfect Numbers are there which are also Super Perfect ?

Details and Assumptions

$n$ is called a superperfect number when $\sigma(\sigma(n))=2n$

$\sigma(n)$ denotes the sum of the divisors of $n$ (from $1$ to $n$ inclusive).

Cant' be determined 2 Infinitely Many 23 0 11 5

1 solution

Kalpok Guha
Mar 13, 2015

$\sigma(\sigma(n))=2n$

or, $\sigma(2n)=2n$ [As $n$ is perfect $\sigma(n)=2n$ ]

Always $\sigma(k)>k$ , as the minimum value of $\sigma(k)$ is $k+1$ , when $k$ is prime.

$\sigma(2n)=2n$ so this cannot be true.So there are no perfect number which is superperfect.So the answer is $\boxed{0}$

Please fix the $\LaTeX$ of the last line.

Prasun Biswas - 6 years, 2 months ago

I think it is fixed

Kalpok Guha - 6 years, 2 months ago

image

Prasun Biswas - 6 years, 2 months ago

@Prasun Biswas – sorry sorry edited it

Kalpok Guha - 6 years, 2 months ago

@Kalpok Guha – Thanks! :)

Prasun Biswas - 6 years, 2 months ago