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 tetrahedral number which is the sum of two tetrahedral numbers:
- $20 = 10 + 10$
- 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 divisor sum which is itself square:
- $\map {\sigma_1} {20^2} = 31^2$
- The $4$th tetrahedral number, after $1$, $4$, $10$:
- $20 = 1 + 3 + 6 + 10 = \dfrac {4 \paren {4 + 1} \paren {4 + 2} } 6$
- The $4$th semiperfect number after $6$, $12$, $18$:
- The $2$nd primitive semiperfect number after $6$:
- $20 = 1 + 4 + 5 + 10$
- The $6$th positive integer after $6$, $9$, $12$, $18$, $19$ whose cube can be expressed as the sum of $3$ positive cube numbers:
- $20^3 = 7^3 + 14^3 + 17^3$
- 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 $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$:
- $\map {\sigma_1} {20} = 42$
- 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 $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $12$, $18$:
- $20 = 10 \times 2 = 10 \times \paren {2 + 0}$
- The $16$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $8$, $9$, $10$, $12$, $13$, $14$, $18$, $19$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
- $20^3 = 11^3 + 12^3 + 13^3 + 14^3$
- 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
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- Previous ... Next: Integers not Expressible as Sum of Distinct Primes of form 6n-1
- Previous ... Next: Numbers not Expressible as Sum of Distinct Pentagonal Numbers
- Previous ... Next: Cubes which are Sum of Three Cubes
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 Divisor Sum is Square/Examples
- Tetrahedral Numbers/Examples
- Harshad Numbers/Examples
- Abundant Numbers/Examples
- Semiperfect Numbers/Examples
- Highly Abundant Numbers/Examples
- Integers not Expressible as Sum of Distinct Primes of form 6n-1/Examples
- Primitive Abundant Numbers/Examples
- Specific Numbers
- 20