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# 14

Please do not rely on any information it contains.

14 is an integer, the smallest even nontotient (there is no solution to ${\displaystyle \phi (n)=14}$, where ${\displaystyle \phi (n)}$ is Euler's totient function).

## Membership in core sequences

 Even numbers ..., 8, 10, 12, 14, 16, 18, 20, ... A005843 Composite numbers ..., 9, 10, 12, 14, 15, 16, 18, ... A002808 Semiprimes 4, 6, 9, 10, 14, 15, 21, 22, 25, ... A001358 Squarefree numbers ..., 10, 11, 13, 14, 15, 17, 19, ... A005117 Square pyramidal numbers 1, 5, 14, 30, 55, 91, 140, 204, ... A000330 Catalan numbers 1, 1, 2, 5, 14, 42, 132, 429, ... A000108

## Sequences pertaining to 14

 Multiples of 14 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, ... A008596 14-gonal numbers 1, 14, 39, 76, 125, 186, 259, 344, 441, 550, 671, 804, ... A051866 Centered 14-gonal numbers 1, 15, 43, 85, 141, 211, 295, 393, 505, 631, 771, 925, ... A069127 ${\displaystyle 3x+1}$ sequence starting at 99 ..., 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, ... A008882 ${\displaystyle 5x+1}$ sequence starting at 11 11, 56, 28, 14, 7, 36, 18, 9, 46, 23, 116, 58, 29, 146, ... A259193

## Partitions of 14

There are 135 partitions of 14.

## Roots and powers of 14

In the table below, irrational numbers are given truncated to eight decimal places.

 ${\displaystyle {\sqrt {14}}}$ 3.74165738 A010471 14 2 196 ${\displaystyle {\sqrt[{3}]{14}}}$ 2.41014226 A010586 14 3 2744 ${\displaystyle {\sqrt[{4}]{14}}}$ 1.93433642 A011011 14 4 38416 ${\displaystyle {\sqrt[{5}]{14}}}$ 1.69521820 A011099 14 5 537824 ${\displaystyle {\sqrt[{6}]{14}}}$ 1.55246328 A011335 14 6 7529536 ${\displaystyle {\sqrt[{7}]{14}}}$ 1.45791624 A011336 14 7 105413504 ${\displaystyle {\sqrt[{8}]{14}}}$ 1.39080423 A011337 14 8 1475789056 ${\displaystyle {\sqrt[{9}]{14}}}$ 1.34074924 A011338 14 9 20661046784 ${\displaystyle {\sqrt[{10}]{14}}}$ 1.30200545 A011339 14 10 289254654976 A001023

## Logarithms and fourteenth powers

In the OEIS specifically and mathematics in general, ${\displaystyle \log x}$ refers to the natural logarithm of ${\displaystyle x}$, whereas all other bases are specified with a subscript.

If ${\displaystyle n}$ is not a multiple of 29, then either ${\displaystyle n^{14}-1}$ or ${\displaystyle n^{14}+1}$ is. Hence the formula for the Legendre symbol ${\displaystyle \left({\frac {a}{29}}\right)=a^{14}\mod 29}$.

As above, irrational numbers in the following table are truncated to eight decimal places.

 ${\displaystyle \log _{14}2}$ 0.26265 A152780 ${\displaystyle \log _{2}14}$ 3.80735 A154462 2 14 16384 ${\displaystyle \log _{14}3}$ 0.41629 ${\displaystyle \log _{3}14}$ 2.40217 3 14 4782969 ${\displaystyle \log _{14}4}$ 0.525299 ${\displaystyle \log _{4}14}$ 1.90368 4 14 268435456 ${\displaystyle \log _{14}5}$ 0.609853 ${\displaystyle \log _{5}14}$ 1.63974 5 14 6103515625 ${\displaystyle \log _{14}6}$ 0.678939 ${\displaystyle \log _{6}14}$ 1.47289 6 14 78364164096 ${\displaystyle \log _{14}7}$ 0.73735 ${\displaystyle \log _{7}14}$ 1.35621 7 14 ${\displaystyle \log _{14}8}$ 0.787949 ${\displaystyle \log _{8}14}$ 1.26912 8 14 ${\displaystyle \log _{14}9}$ 0.832579 ${\displaystyle \log _{9}14}$ 1.20109 9 14 ${\displaystyle \log _{14}10}$ 0.872503 ${\displaystyle \log _{10}14}$ 1.14613 10 14 ${\displaystyle \log _{14}11}$ 0.908618 ${\displaystyle \log _{11}14}$ 1.10057 11 14 ${\displaystyle \log _{14}12}$ 0.941589 ${\displaystyle \log _{12}14}$ 1.06203 12 14 ${\displaystyle \log _{14}13}$ 0.971919 ${\displaystyle \log _{13}14}$ 1.02889 13 14 ${\displaystyle \log _{14}14}$ 1 ${\displaystyle \log _{14}14}$ 1 14 14

(See A010802 for the fourteenth powers of integers).

## Values for number theoretic functions with 14 as an argument

 ${\displaystyle \mu (14)}$ 1 ${\displaystyle M(14)}$ −2 ${\displaystyle \pi (14)}$ 6 ${\displaystyle \sigma _{1}(14)}$ 24 ${\displaystyle \sigma _{0}(14)}$ 4 ${\displaystyle \phi (14)}$ 6 ${\displaystyle \Omega (14)}$ 2 ${\displaystyle \omega (14)}$ 2 ${\displaystyle \lambda (14)}$ 6 This is the Carmichael lambda function. ${\displaystyle \lambda (14)}$ 1 This is the Liouville lambda function. ${\displaystyle \zeta (14)}$ 1.00006124... ${\displaystyle {\frac {2\pi ^{14}}{18243225}}}$ (see A013672). 14! 87178291200 ${\displaystyle \Gamma (14)}$ 6227020800

## Factorization of some small integers in a quadratic integer ring with discriminant −14, 14

The commutative quadratic integer ring with unity ${\displaystyle \scriptstyle \mathbb {Z} [{\sqrt {14}}]\,}$, with units of the form ${\displaystyle \scriptstyle \pm (15+4{\sqrt {14}})^{n}\,}$ (${\displaystyle \scriptstyle n\,\in \,\mathbb {Z} \,}$), is a unique factorization domain. However, it is not norm-Euclidean, and the fact that it's a Euclidean domain was proved only very recently.[1]

${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ is not a unique factorization domain. But in it, there are only two units: 1 and –1. Therefore, we can say with much greater confidence that we have correctly identified instances of multiple factorization.

 ${\displaystyle n}$ ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {14}}]}$ 2 Irreducible ${\displaystyle (4-{\sqrt {14}})(4+{\sqrt {14}})}$ 3 Prime 4 2 2 ${\displaystyle (4-{\sqrt {14}})^{2}(4+{\sqrt {14}})^{2}}$ 5 Irreducible ${\displaystyle (-1)(3-{\sqrt {14}})(3+{\sqrt {14}})}$ 6 2 × 3 ${\displaystyle (4\pm {\sqrt {14}})3}$ 7 Irreducible ${\displaystyle (-1)(7-2{\sqrt {14}})(7+2{\sqrt {14}})}$ 8 2 3 ${\displaystyle (4-{\sqrt {14}})^{3}(4+{\sqrt {14}})^{3}}$ 9 3 2 10 2 × 5 ${\displaystyle (-1)(4\pm {\sqrt {14}})(3\pm {\sqrt {14}})}$ 11 Prime ${\displaystyle (5-{\sqrt {14}})(5+{\sqrt {14}})}$ 12 2 2 × 3 ${\displaystyle (4\pm {\sqrt {14}})^{2}3}$ 13 Irreducible ${\displaystyle (-1)(1-{\sqrt {14}})(1+{\sqrt {14}})}$ 14 2 × 7 OR ${\displaystyle (-1)({\sqrt {-14}})^{2}}$ ${\displaystyle (-1)(4\pm {\sqrt {14}})(7\pm 2{\sqrt {14}})}$ 15 3 × 5 OR ${\displaystyle (1-{\sqrt {-14}})(1+{\sqrt {-14}})}$ ${\displaystyle (-1)3(3\pm {\sqrt {14}})}$ 16 2 4 ${\displaystyle (4\pm {\sqrt {14}})^{4}}$ 17 Prime 18 2 × 3 2 OR ${\displaystyle (2-{\sqrt {-14}})(2+{\sqrt {-14}})}$ ${\displaystyle (4\pm {\sqrt {14}})3^{2}}$ 19 Irreducible Prime 20 2 2 × 5 ${\displaystyle (-1)(4\pm {\sqrt {14}})^{2}(3\pm {\sqrt {14}})}$

The factorization of 14 points up an important difference between ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ and ${\displaystyle \mathbb {Z} [{\sqrt {14}}]}$. In the former, 2, 7 and ${\displaystyle {\sqrt {-14}}}$ are all irreducible. In the latter, ${\displaystyle {\sqrt {14}}}$ is composite, since indeed ${\displaystyle (4-{\sqrt {14}})(7+2{\sqrt {14}})={\sqrt {14}}}$. Also note that the "alternate" factorization of 18 in ${\displaystyle {\sqrt {-14}}}$ has one fewer irreducible factor than the "standard" factorization; ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ has class number 4.

Ideals really help us make sense of multiple distinct factorizations ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$.

 ${\displaystyle p}$ Factorization of ${\displaystyle \langle p\rangle }$ In ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ In ${\displaystyle \mathbb {Z} [{\sqrt {14}}]}$ 2 ${\displaystyle \langle 2,{\sqrt {-14}}\rangle ^{2}}$ ${\displaystyle \langle 4+{\sqrt {14}}\rangle ^{2}}$ 3 ${\displaystyle \langle 3,1-{\sqrt {-14}}\rangle \langle 3,1+{\sqrt {-14}}\rangle }$ Prime 5 ${\displaystyle \langle 5,1-{\sqrt {-14}}\rangle \langle 5,1+{\sqrt {-14}}\rangle }$ ${\displaystyle \langle 3-{\sqrt {14}}\rangle \langle 3+{\sqrt {14}}\rangle }$ 7 ${\displaystyle \langle 7,{\sqrt {-14}}\rangle ^{2}}$ 11 Prime ${\displaystyle \langle 5-{\sqrt {14}}\rangle \langle 5+{\sqrt {14}}\rangle }$ 13 ${\displaystyle \langle 13,5-{\sqrt {-14}}\rangle \langle 13,5+{\sqrt {-14}}\rangle }$ ${\displaystyle \langle 1-{\sqrt {14}}\rangle \langle 1+{\sqrt {14}}\rangle }$ 17 Prime 19 ${\displaystyle \langle 19,9-{\sqrt {-14}}\rangle \langle 19,9+{\sqrt {-14}}\rangle }$ Prime 23 ${\displaystyle \langle 3-{\sqrt {-14}}\rangle \langle 3+{\sqrt {-14}}\rangle }$ 29 Prime 31 Prime 37 41 43 47

## Factorization of 14 in some quadratic integer rings

In ${\displaystyle \mathbb {Z} }$, 14 is the product of 2 and 7. But it has different factorizations in some integer rings.

 ${\displaystyle \mathbb {Z} [i]}$ ${\displaystyle (1-i)(1+i)7}$ ${\displaystyle \mathbb {Z} [{\sqrt {-2}}]}$ ${\displaystyle (-1)({\sqrt {-2}})^{2}7}$ ${\displaystyle \mathbb {Z} [{\sqrt {2}}]}$ ${\displaystyle ({\sqrt {2}})^{2}(3-{\sqrt {2}})(3+{\sqrt {2}})}$ ${\displaystyle \mathbb {Z} [\omega ]}$ 2 × 7 ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$ ${\displaystyle (-1)(1-{\sqrt {3}})(1+{\sqrt {3}})7}$ ${\displaystyle \mathbb {Z} [{\sqrt {-5}}]}$ 2 × 7 OR ${\displaystyle (3-{\sqrt {-5}})(3+{\sqrt {-5}})}$ ${\displaystyle \mathbb {Z} [\phi ]}$ 2 × 7 ${\displaystyle \mathbb {Z} [{\sqrt {-6}}]}$ ${\displaystyle 2(1-{\sqrt {-6}})(1+{\sqrt {-6}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {6}}]}$ ${\displaystyle (-1)(2-{\sqrt {6}})(2+{\sqrt {6}})7}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}}$ ${\displaystyle (-1)\left({\frac {1}{2}}-{\frac {\sqrt {-7}}{2}}\right)\left({\frac {1}{2}}+{\frac {\sqrt {-7}}{2}}\right)({\sqrt {-7}})^{2}}$ ${\displaystyle \mathbb {Z} [{\sqrt {7}}]}$ ${\displaystyle (3-{\sqrt {7}})(3+{\sqrt {7}})({\sqrt {7}})^{2}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-10}}]}$ 2 × 7 OR ${\displaystyle (2-{\sqrt {-10}})(2+{\sqrt {-10}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {10}}]}$ 2 × 7 ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}}$ 2 × 7 ${\displaystyle \mathbb {Z} [{\sqrt {11}}]}$ ${\displaystyle (3-{\sqrt {11}})(3+{\sqrt {11}})(2-{\sqrt {11}})(2+{\sqrt {11}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {-13}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}}$ 2 × 7 ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ 2 × 7 OR ${\displaystyle (-1)({\sqrt {-14}})^{2}}$ ${\displaystyle \mathbb {Z} [{\sqrt {14}}]}$ ${\displaystyle (-1)(4\pm {\sqrt {14}})(7\pm 2{\sqrt {14}})}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}}$ 2 × 7 ${\displaystyle \mathbb {Z} [{\sqrt {15}}]}$ 2 × 7 OR ${\displaystyle (-1)(1-{\sqrt {15}})(1+{\sqrt {15}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {-17}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}}$ ${\displaystyle (-1)\left({\frac {3}{2}}-{\frac {\sqrt {17}}{2}}\right)\left({\frac {3}{2}}+{\frac {\sqrt {17}}{2}}\right)7}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}}$ ${\displaystyle 2\left({\frac {3}{2}}-{\frac {\sqrt {-19}}{2}}\right)\left({\frac {3}{2}}+{\frac {\sqrt {-19}}{2}}\right)}$ ${\displaystyle \mathbb {Z} [{\sqrt {19}}]}$ ${\displaystyle (-1)(13-3{\sqrt {19}})(13+3{\sqrt {19}})7}$

## Representation of 14 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 through 36 Representation 1110 112 32 24 22 20 16 15 14 13 12 11 10 E

 ${\displaystyle -1}$ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1729
1. Malcolm Harper, "${\displaystyle \mathbb {Z} [{\sqrt {14}}]}$ is Euclidean" Canad. J. Math. Vol. 56 (1), 2004 pp. 55-70.