# 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

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $16$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $16$