# 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 $\map \sigma n = \map \sigma {n + 1}$:
$\map \sigma {14} = 24 = \map \sigma {15}$

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 $1$st of the $4$th pair of consecutive integers whose product is a primorial:
$14 \times 15 = 210 = 7 \#$

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 $\tau$ function equals its $\sigma$ function:
$\map \phi {14} \, \map \tau {14} = 6 \times 4 = 24 = \map \sigma {14}$

The $4$th Catalan number after $(1)$, $1$, $2$, $5$:
$\dfrac 1 {4 + 1} \dbinom {2 \times 4} 4 = \dfrac 1 5 \times 70 = 14$

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 $7$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 $12$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:
$2^{14} = 16 \, 384$

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

## Historical Note

The two main occurrences of the number $14$ of contemporary social significance are:

the number of pounds avoirdupois in one stone
the number of days in the fortnight.