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# 6

Please do not rely on any information it contains.

6 is the smallest squarefree semiprime, being the product of 2 and 3.

## Membership in core sequences

 Even numbers 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, ... A005843 Composite numbers 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, ... A002808 Semiprimes 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, ... A001358 Triangular numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, ... A000217 Factorials 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ... A000142

In Pascal's triangle, 6 occurs thrice, the first time with 4 on either side. (In Lozanić's triangle, 6 occurs six times).

## Sequences pertaining to 6

 Multiples of 6 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, ... A008588 Inert rational primes in ${\displaystyle \mathbb {Q} ({\sqrt {6}})}$ 7, 11, 13, 17, 31, 37, 41, 59, 61, 79, 83, 89, 103, 107, ... A038877 Fermat pseudoprimes to base 6 35, 185, 217, 301, 481, 1105, 1111, 1261, 1333, 1729, ... A005937 Hexagonal numbers 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, ... A000384 Primes with primitive root 6 11, 13, 17, 41, 59, 61, 79, 83, 89, 103, 107, 109, 113, ... A019336

## Partitions of 6

There are eleven partitions of 6, with the longest one consisting of distinct numbers being {1, 2, 3}. The only possible prime partitions are {2, 2, 2} and {3, 3}.

## Roots and powers of 6

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

 ${\displaystyle {\sqrt {6}}}$ 2.44948974 A010464 6 2 36 ${\displaystyle {\sqrt[{3}]{6}}}$ 1.81712059 A005486 6 3 216 ${\displaystyle {\sqrt[{4}]{6}}}$ 1.56508458 A011004 6 4 1296 ${\displaystyle {\sqrt[{5}]{6}}}$ 1.43096908 A011091 6 5 7776 ${\displaystyle {\sqrt[{6}]{6}}}$ 1.34800615 A011215 6 6 46656 ${\displaystyle {\sqrt[{7}]{6}}}$ 1.29170834 A011216 6 7 279936 ${\displaystyle {\sqrt[{8}]{6}}}$ 1.25103340 A011217 6 8 1679616 ${\displaystyle {\sqrt[{9}]{6}}}$ 1.22028493 A011218 6 9 10077696 ${\displaystyle {\sqrt[{10}]{6}}}$ 1.19623119 A011219 6 10 60466176 ${\displaystyle {\sqrt[{11}]{6}}}$ 1.17690395 A011220 6 11 362797056 ${\displaystyle {\sqrt[{12}]{6}}}$ 1.16103667 A011221 6 12 2176782336 ${\displaystyle {\sqrt[{13}]{6}}}$ 1.14777771 A011222 6 13 13060694016 ${\displaystyle {\sqrt[{14}]{6}}}$ 1.13653347 A011223 6 14 78364164096 ${\displaystyle {\sqrt[{15}]{6}}}$ 1.12687761 A011224 6 15 470184984576 ${\displaystyle {\sqrt[{16}]{6}}}$ 1.11849604 A011225 6 16 2821109907456 A000400

## Logarithms and sixth 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 sixth powers are both squares and cubes, since ${\displaystyle b^{6}=(b^{3})^{2}=(b^{2})^{3}}$. And from Fermat's little theorem it follows that if ${\displaystyle b}$ is coprime to 7, then ${\displaystyle b^{6}\equiv 1\mod 7}$.

If ${\displaystyle n}$ is not a multiple of 13, then either ${\displaystyle n^{6}-1}$ or ${\displaystyle n^{6}+1}$ is. Hence the formula for the Legendre symbol ${\displaystyle \left({\frac {a}{13}}\right)=a^{6}\mod 13}$.

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

 ${\displaystyle \log _{6}2}$ 0.38685280 A152683 ${\displaystyle \log _{2}6}$ 2.58496250 A020859 2 6 64 ${\displaystyle \log _{6}e}$ 0.55811062 ${\displaystyle \log 6}$ 1.79175946 A016629 ${\displaystyle e^{6}}$ 403.429 A092512 ${\displaystyle \log _{6}3}$ 0.61314719 A152935 ${\displaystyle \log _{3}6}$ 1.63092975 A153459 3 6 729 ${\displaystyle \log _{6}\pi }$ 0.63888591 ${\displaystyle \log _{\pi }6}$ 1.56522467 ${\displaystyle \pi ^{6}}$ 961.389 A092732 ${\displaystyle \log _{6}4}$ 0.77370561 A153102 ${\displaystyle \log _{4}6}$ 1.29248125 A153460 4 6 4096 ${\displaystyle \log _{6}5}$ 0.89824440 A153202 ${\displaystyle \log _{5}6}$ 1.11328275 A153461 5 6 15625 ${\displaystyle \log _{6}6}$ 1.00000000 6 6 46656 ${\displaystyle \log _{6}7}$ 1.08603313 A153617 ${\displaystyle \log _{7}6}$ 0.92078222 A153463 7 6 117649 ${\displaystyle \log _{6}8}$ 1.16055842 A153754 ${\displaystyle \log _{8}6}$ 0.86165416 A153493 8 6 262144 ${\displaystyle \log _{6}9}$ 1.22629438 A154009 ${\displaystyle \log _{9}6}$ 0.81546487 A153495 9 6 531441 ${\displaystyle \log _{6}10}$ 1.28509720 A154157 ${\displaystyle \log _{10}6}$ 0.77815125 A153496 10 6 1e+06

(See A001014 for the sixth powers of integers).

## Values for number theoretic functions with 6 as an argument

 ${\displaystyle \mu (6)}$ 1 ${\displaystyle M(6)}$ –1 ${\displaystyle \pi (6)}$ 3 ${\displaystyle \sigma _{1}(6)}$ 12 Note that this is twice 6, so 6 is a perfect number. ${\displaystyle \sigma _{0}(6)}$ 4 ${\displaystyle \phi (6)}$ 2 This is the largest ${\displaystyle n}$ such that ${\displaystyle \phi (n)<{\sqrt {n}}}$. ${\displaystyle \Omega (6)}$ 2 ${\displaystyle \omega (6)}$ 2 ${\displaystyle \lambda (6)}$ 2 This is the Carmichael lambda function. ${\displaystyle \lambda (6)}$ 1 This is the Liouville lambda function. ${\displaystyle \zeta (6)={\frac {\pi ^{6}}{945}}}$ 1.01734306198444913971451792979... (see A013664) 6! 720 ${\displaystyle \Gamma (6)}$ 120

## Factorization of some small integers in a quadratic integer ring adjoining the square root of −6 or 6

We've got a bit of a contrast here between ${\displaystyle \mathbb {Z} [{\sqrt {-6}}]}$ and ${\displaystyle \mathbb {Z} [{\sqrt {6}}]}$ here: the latter is a unique factorization domain, the former is not. Now, it may seem rather strange that ${\displaystyle ({\sqrt {6}})^{2}}$ is not a distinct factorization of 6 in ${\displaystyle \mathbb {Z} [{\sqrt {6}}]}$. But since that's a UFD, in combination with the fact that neither 2 nor 3 are prime in it, suggests that ${\displaystyle {\sqrt {6}}}$ is actually composite, and indeed we see that ${\displaystyle (2+{\sqrt {6}})(3-{\sqrt {6}})={\sqrt {6}}}$.

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

Note that ${\displaystyle (4-{\sqrt {6}})(4+{\sqrt {6}})=10}$ is not a distinct factorization, since ${\displaystyle (2-{\sqrt {6}})(-1-{\sqrt {6}})=(4-{\sqrt {6}})}$.

Ideals help us make sense of multiple distinct factorizations in ${\displaystyle \mathbb {Z} [{\sqrt {-6}}]}$.

TABLE OF FACTORIZATION OF IDEALS GOES HERE

## Factorization of 6 in some quadratic integer rings

As was mentioned above, 6 is a squarefree semiprime in ${\displaystyle \mathbb {Z} }$. But it has different factorizations in some quadratic integer rings. Often, 6 is the classic example that unique factorization does not hold in a given ring, such as ${\displaystyle \mathbb {Z} [{\sqrt {-5}}]}$: one shows that 2 and 3 are irreducible in that ring, yet 6 can be expressed a product of two other irreducibles "coprime" to 2 and 3.

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

## Representation of 6 in various bases

 Base 2 3 4 5 6 7 through 36 Representation 110 20 12 11 10 6

Note that 6 is palindromic in base 5, but of course this is to be expected of any integer ${\displaystyle n>2}$ in base ${\displaystyle n-1}$. In base 7 and up, 6 is trivially palindromic. But it is not palindromic in binary, ternary or quartal, hence it is called a strictly non-palindromic number (see A016038).

 ${\displaystyle -1}$ 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