18

== Number ==
$18$ (eighteen) is:
:$2 \times 3^2$
:The only number which is twice the sum of its digits:
::$18 = 2 \times \paren {1 + 8}$
:The $1$st element of the $1$st pair of integers $m$ whose values of $m \, \map {\sigma_0} m$ is equal:
::$18 \times \map {\sigma_0} {18} = 108 = 27 \times \map {\sigma_0} {27}$
:The $2$nd abundant number after $12$:
::$1 + 2 + 3 + 6 + 9 = 21 > 18$
:The $3$rd semiperfect number after $6$, $12$:
::$18 = 3 + 6 + 9$
:The $3$rd heptagonal number after $1$, $7$:
::$18 = 1 + 7 + 11 = \dfrac {3 \paren {5 \times 3 - 3} } 2$
:The $3$rd pentagonal pyramidal number after $1$, $6$:
::$18 = 1 + 5 + 12 = \dfrac {3^2 \paren {3 + 1} } 2$
:The $4$th positive integer after $6$, $9$, $12$ whose cube can be expressed as the sum of $3$ positive cube numbers:
::$18^3 = 2^3 + 12^3 + 16^3 = 9^3 + 12^3 + 15^3$
:The number of distinct fixed pentominoes
:The $5$th Dudeney number after $0$, $1$, $8$, $17$:
::$18^3 = 5832$, while $5 + 8 + 3 + 2 = 18$
:Equal to the sum of the digits of its $6$th power:
::$18^6 = 34 \, 012 \, 224$, while $3 + 4 + 0 + 1 + 2 + 2 + 2 + 4 = 18$
:Equal to the sum of the digits of its $7$th power:
::$18^7 = 612 \, 220 \, 032$, while $6 + 1 + 2 + 2 + 2 + 0 + 0 + 3 + 2 = 18$
:The $6$th Lucas number after $(2)$, $1$, $3$, $4$, $7$, $11$:
::$18 = 7 + 11$
:The $8$th positive integer after $1$, $2$, $3$, $4$, $6$, $8$, $12$ such that all smaller positive integers coprime to it are prime
:The $9$th positive integer which is not the sum of $1$ or more distinct squares:
::$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $\ldots$
:The $10$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$:
::$\map {\sigma_1} {18} = 39$
:The $10$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$, $16$:
::$18 = 2 + 16$
:The $10$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $9$ such that both $2^n$ and $5^n$ have no zeroes:
::$2^{18} = 262 \, 144$, $5^{18} = 3 \, 814 \, 697 \, 265 \, 625$
:The $11$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $9$, $10$, $12$, $15$ which cannot be expressed as the sum of exactly $5$ non-zero squares
:The $12$th harshad number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $12$:
::$18 = 2 \times 9 = 2 \times \paren {1 + 8}$
:The $13$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $9$, $10$, $11$, $17$ such that $5^n$ contains no zero in its decimal representation:
::$5^{18} = 3 \, 814 \, 697 \, 265 \, 625$
:The $13$th after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $10$, $12$, $14$, $16$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways
:The $14$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $8$, $9$, $10$, $12$, $13$, $14$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
:The $15$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $13$, $14$, $15$, $16$ such that $2^n$ contains no zero in its decimal representation:
::$2^{18} = 262 \, 144$
:$18 = 9 + 9$, and its reversal $81 = 9 \times 9$
:$18^3 = 5832$ and $18^4 = 104 \, 976$, using all $10$ digits from $0$ to $9$ once each between them
== Also see ==
* 18 Fixed Pentominoes
* Only Number Twice Sum of Digits is 18
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== Sources ==
* : $18$
* : $18$
Category:Specific Numbers
Category:18