# 20

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## Number

$20$ (twenty) is:

$2^2 \times 5$

The $1$st positive integer $n$ such that $6 n + 1$ and $6 n - 1$ are both composite:
$6 \times 20 - 1 = 119 = 7 \times 17$, $6 \times 20 + 1 = 121 = 11^2$

The $1$st primitive abundant number:
The $3$rd abundant number after $12$, $18$:
$1 + 2 + 4 + 5 + 10 = 21 > 20$

The $3$rd central binomial coefficient after $2$, $6$:
$20 = \dbinom {2 \times 3} 3 := \dfrac {6!} {\paren {3!}^2}$

The $3$rd number after $1$, $9$ whose square has a $\sigma$ value which is itself square:
$\sigma \left({20^2}\right) = 31^2$

The $4$th tetrahedral number, after $1$, $4$, $10$:
$20 = 1 + 3 + 6 + 10 = \dfrac {4 \left({4 + 1}\right) \left({4 + 2}\right)} 6$

The $4$th semiperfect number after $6$, $12$, $18$:
The $2$nd primitive semiperfect number after $6$:
$20 = 1 + 4 + 5 + 10$

The $9$th even number after $2$, $4$, $6$, $8$, $10$, $12$, $14$, $16$ which cannot be expressed as the sum of $2$ composite odd numbers.

The $11$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$:
$\sigma \left({20}\right) = 42$

The $13$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$, $17$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.

The $13$th positive integer after $2$, $3$, $4$, $7$, $8$, $9$, $10$, $11$, $14$, $15$, $16$, $19$ which cannot be expressed as the sum of distinct pentagonal numbers.

The $13$th harshad number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $12$, $18$:
$20 = 10 \times 2 = 10 \times \left({2 + 0}\right)$

$20^3 = 11^3 + 12^3 + 13^3 + 14^3$

The number of faces on an icosahedron

The number of vertices on a regular dodecahedron

The number of different ways of playing the first move in chess

## Historical Note

Occurrences of the number $20$ in various cultures in history:

There were $20$ shillings in $1$ pound Sterling in pre-decimal British coinage.
There are $20$ fluid ounces in the imperial pint.

The word score means a set of $20$.