# 14

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## Contents

## 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.

## Also see

*Previous ... Next*: Sequence of Powers of 14*Previous ... Next*: Integers whose Phi times Tau equal Sigma*Previous ... Next*: Representation of 1 as Sum of n Unit Fractions*Previous ... Next*: Square Pyramidal Number*Previous ... Next*: Catalan Number*Previous ... Next*: Consecutive Integers whose Product is Primorial*Previous ... Next*: Semiprime Number*Previous ... Next*: Numbers not Expressible as Sum of Distinct Pentagonal Numbers*Previous ... Next*: Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers*Previous ... Next*: Sequence of Integers whose Factorial minus 1 is Prime*Previous ... Next*: Powers of 2 with no Zero in Decimal Representation

## 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.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): Glossary - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $14$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): Glossary - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $14$