24

== 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 $m \, \map {\sigma_0} m$ is equal:
::$24 \times \map {\sigma_0} {24} = 192 = 32 \times \map {\sigma_0} {32}$
:The $2$nd of three $2$-digit integers divisible by both the sum and product of its digits:
::$24 = \paren {2 + 4} \times 4 = \paren {2 \times 4} \times 3$
:The $2$nd positive integer after $1$ whose Euler $\phi$ value is equal to the product of its digits: 
::$\map \phi {24} = 8 = 2 \times 4$
:$4$ 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 $1$, $2$, $4$, $6$, $12$:
::$\dfrac {\map {\sigma_1} {24} } {24} = \dfrac {60} {24} = 2 \cdotp 5$
:The $6$th trimorphic number after $1$, $4$, $5$, $6$, $9$:
::$24^3 = 13 \, 8 \mathbf {24}$
:The $6$th highly composite number after $1$, $2$, $4$, $6$, $12$:
::$\map {\sigma_0} {24} = 8$
:The $7$th positive integer after $6$, $9$, $12$, $18$, $19$, $20$ whose cube can be expressed as the sum of $3$ positive cube numbers:
::$24^3 = 2^3 + 16^3 + 20^3$
: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 $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$:
::$\map {\sigma_1} {24} = 60$
: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 $14$th after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $10$, $12$, $14$, $16$, $18$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways
: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:
::$2^{24} = 16 \, 777 \, 216$
:The $18$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $8$, $9$, $10$, $12$, $13$, $14$, $18$, $19$, $20$, $21$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
: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:
::$1^2 + 2^2 + \cdots + 24^2 = 70^2$
== Also see ==
* Sum of Squares of Divisors of 24 and 26 are Equal
* Smallest Scalene Obtuse Triangle with Integer Sides and Area
* Smallest Positive Integer with 5 Fibonacci Partitions
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== Historical Note ==
== Sources ==
* : $24$
* : $24$
Category:Specific Numbers
Category:24