Numbers whose Sigma is Square

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Theorem

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

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

This sequence is A006532 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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

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


Also see


Sources