14

Number
$14$ (fourteen) is:


 * $2 \times 7$


 * The $5$th semiprime after $4, 6, 9, 10$:
 * $14 = 2 \times 7$


 * The $3$rd square pyramidal number after $1, 5$:
 * $14 = 1 + 4 + 9 = \dfrac {3 \left({3 + 1}\right) \left({2 \times 3 + 1}\right)} 6$


 * 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 $1$st positive integer solution to $\sigma \left({n}\right) = \sigma \left({n + 1}\right)$:
 * $\sigma \left({14}\right) = 24 = \sigma \left({15}\right)$


 * The smallest nontotient:
 * $\nexists m \in \Z_{>0}: \phi \left({m}\right) = 14$
 * where $\phi \left({m}\right)$ denotes the Euler $\phi$ function


 * The smallest Keith number:
 * $1, 4, 5, 9, 14, \ldots$


 * 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 $3$rd positive integer after $1, 3$ of which the product of its Euler $\phi$ function and its $\tau$ function equals its $\sigma$ function:
 * $\phi \left({14}\right) \tau \left({14}\right) = 6 \times 4 = 24 = \sigma \left({14}\right)$


 * 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.


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

Also see

 * 12 times Sigma of 12 equals 14 times Sigma of 14