Amicable numbers occur in pairs. Each
number is equal to the sum of the
other's aliquot divisors (including 1 but
excluding the number itself).
For example, the smallest such numbers are
220 and 284 which were known to the
Pythagoreans.
The proper divisors of 220 are 1, 2, 4, 5,
10, 11, 20, 22, 44, 55, and 110. These add
up to 284.
The proper divisors of 284 are 1, 2, 4, 71,
and 142. They add up to 220.
The Pythagoreans regarded 220 and 284 as
symbols of friendship.
Biblical commentators noticed that Jacob
gave Esau 220 goats and 220 sheep.
(Genesis 32:14).
The commentators believed it was a wise
choice because 220, being one of the amicable
pair, showed Jacob's great love for Esau.
More than 1,000 pairs of amicable numbers are
known today. Here is a list of the amicable
numbers with 5 digits or less:
| 1 | 220 | 284 |
| 2 | 1,184 | 1,210 |
| 3 | 2,620 | 2,924 |
| 4 | 5,020 | 5,564 |
| 5 | 6,232 | 6,368 |
| 6 | 10,744 | 10,856 |
| 7 | 12,285 | 14,595 |
| 8 | 17,296 | 18,416 |
| 9 | 63,020 | 76,084 |
| 10 | 66,928 | 66,992 |
| 11 | 67,095 | 71,145 |
| 12 | 69,615 | 87,633 |
| 13 | 79,750 | 88,730 |
source: Mathematical Magic Show by Martin Gardner