Numbers whose Sigma is Square

Theorem

The sequence of positive integers whose $\sigma$ value is square starts as follows:

 $\displaystyle \map \sigma 3$ $=$ $\displaystyle 4$ $\displaystyle \map \sigma {22}$ $=$ $\displaystyle 36$ $\displaystyle \map \sigma {66}$ $=$ $\displaystyle 144$ $\displaystyle \map \sigma {70}$ $=$ $\displaystyle 144$ $\displaystyle \map \sigma {81}$ $=$ $\displaystyle 121$ $\displaystyle \map \sigma {94}$ $=$ $\displaystyle 144$ $\displaystyle \map \sigma {115}$ $=$ $\displaystyle 144$ $\displaystyle \map \sigma {119}$ $=$ $\displaystyle 144$ $\displaystyle \map \sigma {170}$ $=$ $\displaystyle 324$

Examples

$\sigma$ of $3$ is Square

$\sigma \left({3}\right) = 4 = 2^2$

$\sigma$ of $22$ is Square

$\sigma \left({22}\right) = 36 = 6^2$

$\sigma$ of $66$ is Square

$\map \sigma {66} = 144 = 12^2$

$\sigma$ of $70$ is Square

$\map \sigma {70} = 144 = 12^2$

$\sigma$ of $81$ is Square

$\map \sigma {81} = 121 = 11^2$

$\sigma$ of $94$ is Square

$\sigma \left({94}\right) = 144 = 12^2$

$\sigma$ of $115$ is Square

$\sigma \left({115}\right) = 144 = 12^2$

$\sigma$ of $119$ is Square

$\sigma \left({119}\right) = 144 = 12^2$

$\sigma$ of $400$ is Square

$\map \sigma {400} = 961 = 31^2$