Quasiperfect Number is Square of Odd Integer

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Theorem

Let $n$ be a quasiperfect number.

Then:

$n = \paren {2 k + 1}^2$

for some $k \in \Z_{>0}$.


That is, a quasiperfect number is the square of an odd integer.


Proof

By definition of quasiperfect number:

$\map \sigma n = 2 n + 1$

where $\map \sigma n$ denotes the $\sigma$ function of $n$.

That is, $\map \sigma n$ is odd.

Then from Sigma Function Odd iff Argument is Square or Twice Square:

$n$ is either square or twice a square.



Sources