14

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14 is an integer, the smallest even nontotient (there is no solution to $\varphi (n)=14$, where $\varphi (n)$ 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
$3x+1$ sequence starting at 99	..., 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, ...	A008882
$5x+1$ 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.

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

Logarithms and fourteenth powers

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

If $n$ is not a multiple of 29, then either $n^{14}-1$ or $n^{14}+1$ is. Hence the formula for the Legendre symbol $\left(\frac{a}{29}\right)=a^{14}mod29$.

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

${\log }_{14}2$	0.26264953	A152780	${\log }_{2}14$	3.80735492	A154462	2 14	16384
${\log }_{14}3$	0.41628966		${\log }_{3}14$	2.40217350		3 14	4782969
${\log }_{14}4$	0.52529907		${\log }_{4}14$	1.90367746		4 14	268435456
${\log }_{14}5$	0.60985333		${\log }_{5}14$	1.63973851		5 14	6103515625
${\log }_{14}6$	0.67893919		${\log }_{6}14$	1.47288594		6 14	78364164096
${\log }_{14}7$	0.73735046		${\log }_{7}14$	1.35620718		7 14
${\log }_{14}8$	0.78794860		${\log }_{8}14$	1.26911830		8 14
${\log }_{14}9$	0.83257932		${\log }_{9}14$	1.20108675		9 14
${\log }_{14}10$	0.87250287		${\log }_{10}14$	1.14612803		10 14
${\log }_{14}11$	0.90861810		${\log }_{11}14$	1.10057238		11 14
${\log }_{14}12$	0.94158873		${\log }_{12}14$	1.06203479		12 14
${\log }_{14}13$	0.97191877		${\log }_{13}14$	1.02889256		13 14
${\log }_{14}14$	1.00000000		${\log }_{14}14$	1.00000000		14 14

(See A010802 for the fourteenth powers of integers).

Values for number theoretic functions with 14 as an argument

$\mu (14)$	1
$M(14)$	−2
$\pi (14)$	6
${\sigma }_1(14)$	24
${\sigma }_0(14)$	4
$\varphi (14)$	6
$\mathrm{\Omega }(14)$	2
$\omega (14)$	2
$\lambda (14)$	6	This is the Carmichael lambda function.
$\lambda (14)$	1	This is the Liouville lambda function.
$\zeta (14)$	1.00006124... $\frac{2{\pi }^{14}}{18243225}$ (see A013672).
14!	87178291200
$\mathrm{\Gamma }(14)$	6227020800

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

The commutative quadratic integer ring with unity $\mathbb{\mathbb{Z}}[\sqrt{14}]$, with units of the form $\pm (15+4\sqrt{14}{)}^n$ ($n\in \mathbb{\mathbb{Z}}$), 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.

$\mathbb{\mathbb{Z}}[\sqrt{-14}]$ 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.

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

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

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

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

Factorization of 14 in some quadratic integer rings

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

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

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

See also

Some integers

							$-1$
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