Friendly Pair

Define

$$\Sigma(n)\equiv {\sigma(n)\over n},$$

where $\sigma(n)$ is the Divisor Function. Then a Pair of distinct numbers $(k, m)$ is a friendly pair (and $k$ is said to be a Friend of $m$) if

$$\Sigma(k)=\Sigma(m).$$

For example, 4320 and 4680 are a friendly pair, since $\sigma(4320)=15120$, $\sigma(4680)=16380$, and

$$\Sigma(4320) \equiv {\textstyle{15120\over 4320}}={\textstyle{7\over 2}}$$

$$\Sigma(4680) \equiv {\textstyle{16380\over 4680}}={\textstyle{7\over 2}}.$$

Numbers which do not have Friends are called Solitary Numbers. Solitary Numbers satisfy $(\sigma(n),n)=1$, where $(a,b)$ is the Greatest Common Divisor of $a$ and $b$.

See also Aliquot Sequence, Friend, Solitary Number

References

Anderson, C. W. and Hickerson, D. Problem 6020. ``Friendly Integers.'' Amer. Math. Monthly 84, 65-66, 1977.