14

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Number

$14$ (fourteen) is:

$2 \times 7$

The $1$st power of $14$ after the zeroth $1$:

$14 = 14^1$

The $1$st positive integer solution to $\map {\sigma_1} n = \map {\sigma_1} {n + 1}$:

$\map {\sigma_1} {14} = 24 = \map {\sigma_1} {15}$

where $\sigma_1$ denotes the divisor sum function

The $1$st nontotient:

$\nexists m \in \Z_{>0}: \map \phi m = 14$

where $\map \phi m$ denotes the Euler $\phi$ function

The $1$st Keith number:

$1$, $4$, $5$, $9$, $14$, $\ldots$

The $2$nd integer after $1$ whose divisor sum divided by its Euler $\phi$ value is a square:

$\dfrac {\map {\sigma_1} {14} } {\map \phi {14} } = \dfrac {24} 6 = 4 = 2^2$

The $3$rd square pyramidal number after $1$, $5$:

$14 = 1 + 4 + 9 = \dfrac {3 \paren {3 + 1} \paren {2 \times 3 + 1} } 6$

The $3$rd positive integer after $1$, $3$ of which the product of its Euler $\phi$ function and its divisor count equals its divisor sum:

$\map \phi {14} \map {\sigma_0} {14} = 6 \times 4 = 24 = \map {\sigma_1} {14}$

The $1$st of the $4$th pair of consecutive integers whose product is a primorial:

$14 \times 15 = 210 = 7 \#$

The $4$th Catalan number after $(1)$, $1$, $2$, $5$:

$14 = \dfrac 1 {4 + 1} \dbinom {2 \times 4} 4 = \dfrac 1 5 \times 70$

The number of different representations of $1$ as the sum of $4$ unit fractions.

The $5$th semiprime after $4$, $6$, $9$, $10$:

$14 = 2 \times 7$

The $6$th integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:

$3$, $4$, $6$, $7$, $12$, $14$

The $6$th and last after $1$, $2$, $5$, $6$, $10$ of $6$ integers $n$ such that the alternating group $A_n$ is ambivalent

The $7$th even number after $2$, $4$, $6$, $8$, $10$, $12$ which cannot be expressed as the sum of $2$ composite odd numbers.

The $9$th positive integer after $2$, $3$, $4$, $7$, $8$, $9$, $10$, $11$ which cannot be expressed as the sum of distinct pentagonal numbers.

The $11$th after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $10$, $12$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways

The $12$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $8$, $9$, $10$, $12$, $13$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$

The $12$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $13$ such that $2^n$ contains no zero in its decimal representation:

$2^{14} = 16 \, 384$

There exist $14$ distinct Bravais lattices.

Kuratowski's Closure-Complement Problem

Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$ be a subset of $T$.

By successive applications of the operations of complement relative to $S$ and the closure, there can be as many as $14$ distinct subsets of $S$ (including $A$ itself).