# 220

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## Number

$220$ (two hundred and twenty) is:

$2^2 \times 5 \times 11$

The smaller of the $1$st amicable pair, with $284$:
$\map {\sigma_1} {220} = \map {\sigma_1} {284} = 504 = 220 + 284$
which is also the $1$st Thabit pair:
$220 = 2^2 \times 11 \times 5 = 2^2 \paren {3 \times 2^2 - 1} \paren {3 \times 2^{2 - 1} - 1}$, $284 = 2^2 \paren {9 \times 2^{2 \times 2 - 1} - 1}$

The $10$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $56$, $84$, $120$, $165$:
$220 = 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 = \dfrac {10 \paren {10 + 1} \paren {10 + 2} } 6$

### Arithmetic Functions on $220$

 $\ds \map {\sigma_1} { 220 }$ $=$ $\ds 504$ $\sigma_1$ of $220$

## Historical Note

The numbers $220$ and $284$, being the smallest amicable pair, were traditionally used in conjunction with overtures of friendship.

The number $220$ appears in the Bible as the number of goats that Jacob gave to Esau on the event of their reunion.

It suggests that they may have been known about considerably earlier than the time of Pythagoras of Samos.