12

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Number

$12$ (twelve) is:

$2^2 \times 3$

The base of the duodecimal number system

The number of edges of the regular octahedron and its dual, the cube

The number of distinct pentominoes, up to reflection

The $1$st power of $12$ after the zeroth $1$:

$12 = 12^1$

The $1$st of three $2$-digit integers divisible by both the sum and product of its digits:

$12 = \paren {1 + 2} \times 4 = \paren {1 \times 2} \times 6$

The $1$st abundant number:

$1 + 2 + 3 + 4 + 6 = 16 > 12$

The $2$nd semiperfect number after $6$:

$12 = 2 + 4 + 6$

The $3$rd pentagonal number after $1$, $5$:

$12 = 1 + 4 + 7 = \dfrac {3 \paren {3 \times 3 - 1} } 2$

The $3$rd superfactorial after $1$, $2$:

$12 = 3\$ = 3! \times 2! \times 1!$

The $3$rd positive integer after $6$, $9$ whose cube can be expressed as the sum of $3$ positive cube numbers:

$12^3 = 6^3 + 8^3 + 10^3$

The $4$th special highly composite number after $1$, $2$, $6$

The $4$th of $6$ integers after $2$, $5$, $8$ which cannot be expressed as the sum of distinct triangular numbers

The $5$th highly composite number after $1$, $2$, $4$, $6$:

$\map {\sigma_0} {12} = 6$

The $5$th superabundant number after $1$, $2$, $4$, $6$, and the smallest which is also abundant:

$\dfrac {\map {\sigma_1} {12} } {12} = \dfrac {28} {12} = 2 \cdotp \dot 3$

The $5$th integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:

$3$, $4$, $6$, $7$, $12$

The $5$th generalized pentagonal number after $1$, $2$, $5$, $7$:

$12 = \dfrac {3 \paren {3 \times 3 - 1} } 2$

The $5$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:

$3$, $4$, $9$, $10$, $12$, $\ldots$

The number of distinct free pentominoes

The $6$th even number after $2$, $4$, $6$, $8$, $10$ which cannot be expressed as the sum of $2$ composite odd numbers

The $7$th positive integer after $1$, $2$, $3$, $4$, $6$, $8$ such that all smaller positive integers coprime to it are prime

The $7$th positive integer which is not the sum of $1$ or more distinct squares:

$2$, $3$, $6$, $7$, $8$, $11$, $12$, $\ldots$

The $8$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$:

$\map {\sigma_1} {12} = 28$

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

The $9$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $9$, $10$ which cannot be expressed as the sum of exactly $5$ non-zero squares.

The $10$th after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $10$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways

The $10$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $8$, $9$, $10$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$

The $11$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$:

$12 = 6 \times 2 = 6 \times \paren {1 \times 2}$

The $11$th harshad number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$:

$12 = 4 \times 3 = 4 \times \paren {1 + 2}$

The square of the reversal of $12$ equals the reversal of the square of $12$:

$12^2 = 144$

$21^2 = 441$

$12 = 3 \times 4$, and $56 = 7 \times 8$