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Friendly Pair


\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


For example, 4320 and 4680 are a friendly pair, since $\sigma(4320)=15120$, $\sigma(4680)=16380$, and
$\displaystyle \Sigma(4320)$ $\textstyle \equiv$ $\displaystyle {\textstyle{15120\over 4320}}={\textstyle{7\over 2}}$  
$\displaystyle \Sigma(4680)$ $\textstyle \equiv$ $\displaystyle {\textstyle{16380\over 4680}}={\textstyle{7\over 2}}.$  

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


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

© 1996-9 Eric W. Weisstein