# Numbers whose Sigma is Square

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## Contents

## 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

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $66$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $22$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $66$