# 24

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## Number

$24$ (twenty-four) is:

$2^3 \times 3$

The $1$st positive integer which can be partitioned into distinct Fibonacci numbers in $5$ different ways:
$24 = 21 + 3 = 21 + 2 + 1 = 13 + 8 + 3 = 13 + 8 + 2 + 1 = 13 + 5 + 3 + 2 + 1$

The $1$st composite number the product of whose proper divisors form its cube:
$1 \times 2 \times 3 \times 4 \times 6 \times 8 \times 12 = 24^3$

The $1$st of the $3$ integers which can be expressed as the product of both $3$ and $4$ consecutive integers:
$24 = 2 \times 3 \times 4 = 1 \times 2 \times 3 \times 4$

The area and perimeter of the $1$st of the only $2$ Pythagorean triples which define a Pythagorean triangle whose area equals its perimeter:
$\tuple {6, 8, 10}$

The $1$st element of the $2$nd pair of integers $m$ whose values of $m \, \map \tau m$ is equal:
$24 \times \map \tau {24} = 192 = 32 \times \map \tau {32}$

The $2$nd of three $2$-digit integers divisible by both the sum and product of its digits:
$24 = \paren {2 + 4} \times 4 = \paren {2 \times 4} \times 3$

$4$ factorial:
$24 = 4! = 4 \times 3 \times 2 \times 1$

The $4$th and last integer after $1$, $22$, $23$ which equals the number of digits in its factorial:
$24! = 620 \, 448 \, 401 \, 733 \, 239 \, 439 \, 360 \, 000$
which has $24$ digits

The $4$th abundant number after $12$, $18$, $20$:
$1 + 2 + 3 + 4 + 6 + 8 + 12 = 36 > 24$

The $5$th semiperfect number after $6$, $12$, $18$, $20$:
$24 = 1 + 3 + 8 + 12$

The $5$th integer $m$ after $0$, $1$, $2$, $8$ such that $m^2 = \dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3$ for integer $n$:
$24^2 = \dbinom {15} 0 + \dbinom {15} 1 + \dbinom {15} 2 + \dbinom {15} 3$

The $6$th superabundant number after $1$, $2$, $4$, $6$, $12$:
$\dfrac {\map \sigma {24} } {24} = \dfrac {60} {24} = 2 \cdotp 5$

The $6$th trimorphic number after $1$, $4$, $5$, $6$, $9$:
$24^3 = 13 \, 8 \mathbf {24}$

The $6$th highly composite number after $1$, $2$, $4$, $6$, $12$:
$\map \tau {24} = 8$

The $9$th positive integer after $1$, $2$, $3$, $4$, $6$, $8$, $12$, $18$ such that all smaller positive integers coprime to it are prime

The $12$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$:
$\sigma \left({24}\right) = 60$

The $13$th positive integer which is not the sum of $1$ or more distinct squares:
$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $\ldots$

The $13$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$:
$24 = 3 \times 8 = 3 \times \paren {2 \times 4}$

The $14$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$, $17$, $20$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes

The $15$th harshad number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $12$, $18$, $20$, $21$:
$24 = 4 \times 6 = 4 \times \paren {2 + 4}$

The $15$th positive integer after $2$, $3$, $4$, $7$, $8$, $9$, $10$, $11$, $14$, $15$, $16$, $19$, $20$, $21$ which cannot be expressed as the sum of distinct pentagonal numbers

The $17$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $13$, $14$, $15$, $16$, $18$, $19$ such that $2^n$ contains no zero in its decimal representation:
$2^{24} = 16 \, 777 \, 216$

Divisible by both the sum and product of its digits:
$24 = 4 \times \paren {2 + 4} = 3 \times \paren {2 \times 4}$

The area of the smallest scalene obtuse triangle with integer sides and area:
$24 = \sqrt {16 \paren {16 - 4} \paren {16 - 13} \paren {16 - 15} }$
where $16 = \dfrac{4 + 13 + 15} 2$

The $24$th pyramidal number is a square:
$1^2 + 2^2 + \cdots + 24^2 = 70^2$

### Arithmetic Functions on $24$

 $\displaystyle \map \tau { 24 }$ $=$ $\displaystyle 8$ $\tau$ of $24$ $\displaystyle \map \sigma { 24 }$ $=$ $\displaystyle 60$ $\sigma$ of $24$

## Historical Note

$24$ occurs frequently as a subdivision of a standard measure:

There are $24$ hours in $1$ day.
There are $24$ scruples in $1$ apothecaries' ounce.
There are $24$ grains in $1$ pennyweight.