# 20

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## Contents

## 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

## Also see

- Smallest n such that 6 n + 1 and 6 n - 1 are both Composite
- Cube of 20 is Sum of Sequence of 4 Consecutive Cubes

*Previous ... Next*: Harshad Number*Previous ... Next*: Abundant Number*Previous ... Next*: Semiperfect Number*Previous ... Next*: Highly Abundant Number

## 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 $20$.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $20$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $20$

Categories:

- Primitive Semiperfect Numbers/Examples
- Central Binomial Coefficients/Examples
- Square Numbers whose Sigma is Square/Examples
- Tetrahedral Numbers/Examples
- Harshad Numbers/Examples
- Abundant Numbers/Examples
- Semiperfect Numbers/Examples
- Highly Abundant Numbers/Examples
- Primitive Abundant Numbers/Examples
- Specific Numbers
- 20