Amicable Pair/Examples/220-284

Example of Amicable Pair

$220$ and $284$ are the smallest amicable pair:

$\map {\sigma_1} {220} = \map {\sigma_1} {284} = 504 = 220 + 284$

Let $\map s n$ denote the aliquot sum of $n$.

By definition:

$\map s n = \map {\sigma_1} n - n$

where $\sigma_1$ denotes the divisor sum function.

Thus:

$\ds \map s {220}$

$=$

$\ds \map {\sigma_1} {220} - 220$

$\ds $

$=$

$\ds 504 - 220$

$\ds $

$=$

$\ds 284$

$\ds \map s {284}$

$=$

$\ds \map {\sigma_1} {284} - 284$

$\ds $

$=$

$\ds 504 - 284$

$\ds $

$=$

$\ds 220$

It can be determined by inspection of the aliquot sums of all smaller integers that there is no smaller amicable pair.

$\blacksquare$

The amicable pair $220$ and $284$ were, according to Iamblichus Chalcidensis, known to Pythagoras of Samos.

However, it is strongly supposed by some commentators that they were known even further back than that.

1919: Leonard Eugene Dickson: History of the Theory of Numbers: Volume $\text { I }$ ... (previous) ... (next): Preface
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $220$
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): amicable numbers
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $220$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): amicable numbers
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): amicable numbers
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): amicable numbers
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): amicable numbers

Weisstein, Eric W. "Amicable Pair." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AmicablePair.html