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# 12

Please do not rely on any information it contains.

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.

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

## Logarithms and twelfth 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.

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 ${\displaystyle b}$ is coprime to 13, then ${\displaystyle b^{12}\equiv 1\mod 13}$.

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

 ${\displaystyle \log _{12}2}$ 0.27894294 A152778 ${\displaystyle \log _{2}12}$ 3.58496250 A020864 2 12 4096 ${\displaystyle \log _{12}e}$ 0.40242960 ${\displaystyle \log 12}$ 2.48490664 A016635 ${\displaystyle e^{12}}$ 162755 ${\displaystyle \log _{12}3}$ 0.44211410 A153015 ${\displaystyle \log _{3}12}$ 2.26185950 A154196 3 12 531441 ${\displaystyle \log _{12}\pi }$ 0.46067319 ${\displaystyle \log _{\pi }12}$ 2.17073624 ${\displaystyle \pi ^{12}}$ 924269 ${\displaystyle \log _{12}4}$ 0.55788589 A153105 ${\displaystyle \log _{4}12}$ 1.79248125 A154197 4 12 1.67772e+07 ${\displaystyle \log _{12}5}$ 0.64768546 A153306 ${\displaystyle \log _{5}12}$ 1.54395931 A154198 5 12 2.44141e+08 ${\displaystyle \log _{12}6}$ 0.72105705 A153589 ${\displaystyle \log _{6}12}$ 1.38685280 A154199 6 12 2.17678e+09 ${\displaystyle \log _{12}7}$ 0.78309185 A153622 ${\displaystyle \log _{7}12}$ 1.27698940 A154200 7 12 1.38413e+10 ${\displaystyle \log _{12}8}$ 0.83682883 A153813 ${\displaystyle \log _{8}12}$ 1.19498750 A154201 8 12 6.87195e+10 ${\displaystyle \log _{12}9}$ 0.88422821 A154012 ${\displaystyle \log _{9}12}$ 1.13092975 A154202 9 12 2.8243e+11 ${\displaystyle \log _{12}10}$ 0.92662840 A154162 ${\displaystyle \log _{10}12}$ 1.07918124 A154203 10 12 1e+12 ${\displaystyle \log _{12}11}$ 0.96498404 A154183 ${\displaystyle \log _{11}12}$ 1.03628656 A154204 11 12 3.13843e+12 ${\displaystyle \log _{12}12}$ 1.00000000 12 12 8.9161e+12

(See A008456 for the twelfth powers of integers).

## Values for number theoretic functions with 12 as an argument

 ${\displaystyle \mu (12)}$ 0 ${\displaystyle M(12)}$ −2 ${\displaystyle \pi (12)}$ 5 ${\displaystyle \sigma _{1}(12)}$ 28 ${\displaystyle \sigma _{0}(12)}$ 6 ${\displaystyle \phi (12)}$ 4 ${\displaystyle \Omega (12)}$ 3 ${\displaystyle \omega (12)}$ 2 ${\displaystyle \lambda (12)}$ 2 This is the Carmichael lambda function. ${\displaystyle \lambda (12)}$ −1 This is the Liouville lambda function. ${\displaystyle \zeta (12)}$ 1.00024608... ${\displaystyle {\frac {691\pi ^{12}}{638512875}}}$. See A013670. 12! 479001600 ${\displaystyle \Gamma (12)}$ 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 ${\displaystyle {\sqrt {12}}=2{\sqrt {3}}}$, ${\displaystyle \mathbb {Z} [{\sqrt {12}}]}$ is not integrally closed, being a subdomain of ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$. The situation with ${\displaystyle \mathbb {Z} [{\sqrt {-12}}]}$ has another wrinkle, since ${\displaystyle \mathbb {Z} [{\sqrt {-3}}]}$ is not integrally closed either, so both ${\displaystyle \mathbb {Z} [{\sqrt {-12}}]}$ and ${\displaystyle \mathbb {Z} [{\sqrt {-3}}]}$ are subdomains of ${\displaystyle \mathbb {Z} [\omega ]}$, with ${\displaystyle \omega }$ being a complex cubic root of 1. Despite this, ${\displaystyle \mathbb {Z} [\omega ]}$ is still a quadratic integer ring.

Since 12 is cubefree, we can consider an integrally closed domain of ${\displaystyle \mathbb {Z} [{\sqrt[{3}]{12}}]}$, 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.

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

## 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.

 ${\displaystyle -1}$ 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