12

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12 is an integer.

Membership in core sequences

Even numbers	...,6, 8, 10, 12, 14, 16, 18, ...	A005843
Composite numbers	..., 8, 9, 10, 12, 14, 15, 16, ...	A002808
Abundant numbers	12, 18, 20, 24, 30, 36, 40, 42, ...	A005101
Pell numbers	1, 2, 5, 12, 29, 70, 169, 408, ...	A000129
Oblong numbers	2, 6, 12, 20, 30, 42, 56, 72, 90, ...	A002378

Sequences pertaining to 12

Multiples of 12	12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, ...	A008594
Dodecagonal numbers	1, 12, 33, 64, 105, 156, 217, 288, 369, 460, 561, 672, ...	A051624

Partitions of 12

There are 77 partitions of 12. Of these, there is only one Goldbach partition: 5 + 7.

Roots and powers of 12

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

${\sqrt {12}}$	3.46410161	A010469	12 2	144
${\sqrt[{3}]{12}}$	2.28942848	A010584	12 3	1728
${\sqrt[{4}]{12}}$	1.86120971	A011009	12 4	20736
${\sqrt[{5}]{12}}$	1.64375182	A011097	12 5	248832
${\sqrt[{6}]{12}}$	1.51308574	A011305	12 6	2985984
${\sqrt[{7}]{12}}$	1.42616163	A011306	12 7	35831808
${\sqrt[{8}]{12}}$	1.36426160	A011307	12 8	429981696
${\sqrt[{9}]{12}}$	1.31798062	A011308	12 9	5159780352
${\sqrt[{10}]{12}}$	1.28208885	A011309	12 10	61917364224
${\sqrt[{11}]{12}}$	1.25345107	A011310	12 11	743008370688
${\sqrt[{12}]{12}}$	1.23007550	A011311	12 12	8916100448256
				A001021

Logarithms and twelfth 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.

From the basic properties of exponentiation, it follows that all twelfth powers are squares, cubes, fourth powers and sixth powers. And from Fermat's little theorem it follows that if $b$ is coprime to 13, then $b^{12}\equiv 1\mod 13$.

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

$\log _{12}2$	0.27894294	A152778	$\log _{2}12$	3.58496250	A020864	2 12	4096
$\log _{12}e$	0.40242960		$\log 12$	2.48490664	A016635	$e^{12}$	162754.79141900
$\log _{12}3$	0.44211410	A153015	$\log _{3}12$	2.26185950	A154196	3 12	531441
$\log _{12}\pi$	0.46067319		$\log _{\pi }12$	2.17073624		$\pi ^{12}$	924269.18152337
$\log _{12}4$	0.55788589	A153105	$\log _{4}12$	1.79248125	A154197	4 12	16777216
$\log _{12}5$	0.64768546	A153306	$\log _{5}12$	1.54395931	A154198	5 12	244140625
$\log _{12}6$	0.72105705	A153589	$\log _{6}12$	1.38685280	A154199	6 12	2176782336
$\log _{12}7$	0.78309185	A153622	$\log _{7}12$	1.27698940	A154200	7 12	13841287201
$\log _{12}8$	0.83682883	A153813	$\log _{8}12$	1.19498750	A154201	8 12	68719476736
$\log _{12}9$	0.88422821	A154012	$\log _{9}12$	1.13092975	A154202	9 12	282429536481
$\log _{12}10$	0.92662840	A154162	$\log _{10}12$	1.07918124	A154203	10 12	1000000000000
$\log _{12}11$	0.96498404	A154183	$\log _{11}12$	1.03628656	A154204	11 12	3138428376721
$\log _{12}12$			1.00000000			12 12	8916100448256

(See A008456 for the twelfth powers of integers).

Values for number theoretic functions with 12 as an argument

$\mu (12)$	0
$M(12)$	−2
$\pi (12)$	5
$\sigma _{1}(12)$	28
$\sigma _{0}(12)$	6
$\phi (12)$	4
$\Omega (12)$	3
$\omega (12)$	2
$\lambda (12)$	2	This is the Carmichael lambda function.
$\lambda (12)$	−1	This is the Liouville lambda function.
$\zeta (12)$	1.00024608... ${\frac {691\pi ^{12}}{638512875}}$. See A013670.
12!	479001600
$\Gamma (12)$	39916800

Factorization of some small integers in a cubic integer ring adjoining the cubic root of 12

Clearly 12 is not a perfect square, but it's not squarefree either. Since ${\sqrt {12}}=2{\sqrt {3}}$, $\mathbb {Z} [{\sqrt {12}}]$ is not integrally closed, being a subdomain of $\mathbb {Z} [{\sqrt {3}}]$. The situation with $\mathbb {Z} [{\sqrt {-12}}]$ has another wrinkle, since $\mathbb {Z} [{\sqrt {-3}}]$ is not integrally closed either, so both $\mathbb {Z} [{\sqrt {-12}}]$ and $\mathbb {Z} [{\sqrt {-3}}]$ are subdomains of $\mathbb {Z} [\omega ]$, with $\omega$ being a complex cubic root of 1. Despite this, $\mathbb {Z} [\omega ]$ is still a quadratic integer ring.

Since 12 is cubefree, we can consider an integrally closed domain of $\mathbb {Z} [{\sqrt[{3}]{12}}]$, which is the of the [FINISH WRITING]

Factorization of 12 in some quadratic integer rings

As was mentioned above, 12 is the product of 2 2 and 3. But it has different factorizations in some quadratic integer rings.

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

Representation of 12 in various bases

Base	2	3	4	5	6	7	8	9	10	11	12	13 through 36
Representation	1100	110	30	22	20	15	14	13	12	11	10	C

Notice that 12 is a Harshad number in every base from binary to duodecimal, with the exception of octal.

See also

• 2^n mod n

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