6

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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 $\mathbb{\mathbb{Q}}(\sqrt{6})$	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.

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

Logarithms and sixth 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 sixth powers are both squares and cubes, since $b^6=(b^3{)}^2=(b^2{)}^3$. And from Fermat's little theorem it follows that if $b$ is coprime to 7, then $b^6\equiv 1mod7$.

If $n$ is not a multiple of 13, then either $n^6-1$ or $n^6+1$ is. Hence the formula for the Legendre symbol $\left(\frac{a}{13}\right)=a^{6}mod13$.

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

${\log }_{6}2$	0.38685280	A152683	${\log }_{2}6$	2.58496250	A020859	2 6	64
${\log }_{6}e$	0.55811062		$\log 6$	1.79175946	A016629	$e^6$	403.42879349	A092512
${\log }_{6}3$	0.61314719	A152935	${\log }_{3}6$	1.63092975	A153459	3 6	729
${\log }_6\pi$	0.63888591		${\log }_{\pi }6$	1.56522467		${\pi }^6$	961.38919357	A092732
${\log }_{6}4$	0.77370561	A153102	${\log }_{4}6$	1.29248125	A153460	4 6	4096
${\log }_{6}5$	0.89824440	A153202	${\log }_{5}6$	1.11328275	A153461	5 6	15625
${\log }_{6}6$			1.00000000			6 6	46656
${\log }_{6}7$	1.08603313	A153617	${\log }_{7}6$	0.92078222	A153463	7 6	117649
${\log }_{6}8$	1.16055842	A153754	${\log }_{8}6$	0.86165416	A153493	8 6	262144
${\log }_{6}9$	1.22629438	A154009	${\log }_{9}6$	0.81546487	A153495	9 6	531441
${\log }_{6}10$	1.28509720	A154157	${\log }_{10}6$	0.77815125	A153496	10 6	1000000

(See A001014 for the sixth powers of integers).

Values for number theoretic functions with 6 as an argument

$\mu (6)$	1
$M(6)$	–1
$\pi (6)$	3
${\sigma }_1(6)$	12	Note that this is twice 6, so 6 is a perfect number.
${\sigma }_0(6)$	4
$\varphi (6)$	2	This is the largest $n$ such that $\varphi (n)<\sqrt{n}$.
$\mathrm{\Omega }(6)$	2
$\omega (6)$	2
$\lambda (6)$	2	This is the Carmichael lambda function.
$\lambda (6)$	1	This is the Liouville lambda function.
$\zeta (6)=\frac{{\pi }^6}{945}$	1.01734306198444913971451792979... (see A013664)
6!	720
$\mathrm{\Gamma }(6)$	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 $\mathbb{\mathbb{Z}}[\sqrt{-6}]$ and $\mathbb{\mathbb{Z}}[\sqrt{6}]$ here: the latter is a unique factorization domain, the former is not. Now, it may seem rather strange that $(\sqrt{6}{)}^2$ is not a distinct factorization of 6 in $\mathbb{\mathbb{Z}}[\sqrt{6}]$. But since that's a UFD, in combination with the fact that neither 2 nor 3 are prime in it, suggests that $\sqrt{6}$ is actually composite, and indeed we see that $(2+\sqrt{6})(3-\sqrt{6})=\sqrt{6}$.

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

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

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

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 $\mathbb{\mathbb{Z}}$. 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 $\mathbb{\mathbb{Z}}[\sqrt{-5}]$: 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.

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

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 $n>2$ in base $n-1$. 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).

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