22

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22 is an integer.

Membership in core sequences

Even numbers	..., 14, 16, 18, 20, 22, 24, 26, 28, 30, ...	A005843(11)
Composite numbers	..., 18, 20, 21, 22, 24, 25, 26, 27, 28, ...	A002808
Semiprimes	..., 14 15, 21, 22, 25, 26, 33, 34, 35, ...	A001358
Squarefree numbers	..., 17, 19, 21, 22, 23, 26, 29, 30, 31, ...	A005117
Pentagonal numbers	1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...	A000326

In Pascal's triangle, 22 occurs twice.

Sequences pertaining to 22

Multiples of 22	0, 22, 44, 66, 88, 110, 132, 154, 176, 198, ...	A005843
22-gonal numbers	0, 1, 22, 63, 124, 205, 306, 427, 568, 729, ...	A051874
Centered 22-gonal numbers	1, 23, 67, 133, 221, 331, 463, 617, 793, ...	A069173
Concentric 22-gonal numbers	1, 22, 45, 88, 133, 198, 265, 352, 441, 550, ...	A195149
$3x+1$ sequence beginning at 9	9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, ...	A033479

Partitions of 22

There are 1002 partitions of 22.

The Goldbach representations of 22 are: 3 + 19 = 5 + 17 = 11 + 11.

Roots and powers of 22

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

${\sqrt {22}}$	4.69041575	A010478	22 2	484
${\sqrt[{3}]{22}}$	2.80203933	A010594	22 3	10648
${\sqrt[{4}]{22}}$	2.16573677	A011018	22 4	234256
${\sqrt[{5}]{22}}$	1.85560073	A011107	22 5	5153632
${\sqrt[{6}]{22}}$	1.67392930		22 6	113379904
${\sqrt[{7}]{22}}$	1.55515853		22 7	2494357888
${\sqrt[{8}]{22}}$	1.47164424		22 8	54875873536
${\sqrt[{9}]{22}}$	1.40980184		22 9	1207269217792
${\sqrt[{10}]{22}}$	1.36220436		22 10	26559922791424
				A009966

Logarithms and 22nd 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.

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

TABLE

Values for number theoretic functions with 22 as an argument

$\mu (22)$	1
$M(22)$	–3
$\pi (22)$	8
$\sigma _{1}(22)$	36
$\sigma _{0}(22)$	4
$\phi (22)$	10
$\Omega (22)$	2
$\omega (22)$	2
$\lambda (22)$	10	This is the Carmichael lambda function.
$\lambda (22)$	1	This is the Liouville lambda function.
$\zeta (22)$	1.00000023845... (see A013668).
22!	1124000727777607680000
$\Gamma (22)$	51090942171709440000

Factorization of some small integers in a quadratic integer ring adjoining the square roots of −22, 22

The commutative quadratic integer ring with unity $\scriptstyle \mathbb {Z} [{\sqrt {22}}]\,$, with units of the form $\scriptstyle \pm (197+42{\sqrt {22}})^{n}\,$ ($\scriptstyle n\,\in \,\mathbb {Z} \,$), is a unique factorization domain.

$n$	$\mathbb {Z} [{\sqrt {-22}}]$	$\mathbb {Z} [{\sqrt {22}}]$
2	Irreducible	$(-1)(14-3{\sqrt {22}})(14+3{\sqrt {22}})$
3	Prime	$(5-{\sqrt {22}})(5+{\sqrt {22}})$
4	2 2	$(14\pm 3{\sqrt {22}})^{2}$
5	Prime
6	2 × 3	$(-1)(14\pm 3{\sqrt {22}})(5\pm {\sqrt {22}})$
7	Prime	$(-1)(9-2{\sqrt {22}})(9+2{\sqrt {22}})$
8	2 3	$(-1)(14\pm 3{\sqrt {22}})^{3}$
9	3 2	$(5\pm {\sqrt {22}})^{2}$
10	2 × 5	$(-1)(14\pm 3{\sqrt {22}})5$
11	Irreducible	$(33-7{\sqrt {22}})(33+7{\sqrt {22}})$
12	2 2 × 3	$(14\pm 3{\sqrt {22}})^{2}(5\pm {\sqrt {22}})$
13	Irreducible	$(-1)(3-{\sqrt {22}})(3+{\sqrt {22}})$
14	2 × 7	$(14\pm 3{\sqrt {22}})(9\pm 2{\sqrt {22}})$
15	3 × 5	$(5\pm {\sqrt {22}})5$
16	2 4	$(14\pm 3{\sqrt {22}})^{4}$
17	Prime
18	2 × 3 2	$(-1)(14\pm 3{\sqrt {22}})(5\pm {\sqrt {22}})^{2}$
19	Irreducible	Prime
20	2 2 × 5	$(14\pm 3{\sqrt {22}})^{2}5$
21	3 × 7
22	2 × 11 OR $(-1)({\sqrt {-22}})^{2}$
23	$(1-{\sqrt {-22}})(1+{\sqrt {-22}})$	Prime
24
25
26	2 × 13 OR $(2-{\sqrt {-22}})(2+{\sqrt {-22}})$
27
28
29		$(-1)(13-3{\sqrt {22}})(13+3{\sqrt {22}})$
30

It almost goes without saying that $({\sqrt {22}})^{2}$ is not a distinct factorization of 22, since this is a UFD and we readily see that $(-14-3{\sqrt {22}})(-33+7{\sqrt {22}})={\sqrt {22}}$.

Ideals help us make sense of multiple distinct factorizations.

$p$	Factorization of $\langle p\rangle$
	In $\mathbb {Z} [{\sqrt {-22}}]$	In $\mathbb {Z} [{\sqrt {22}}]$
2	$\langle 2,{\sqrt {-22}}\rangle ^{2}$	$\langle 14+3{\sqrt {22}}\rangle ^{2}$
3	Prime	$\langle 5-{\sqrt {22}}\rangle \langle 5+{\sqrt {22}}\rangle$
5		Prime
7		$\langle 9-2{\sqrt {22}}\rangle \langle 9+2{\sqrt {22}}\rangle$
11	$\langle 11,{\sqrt {-22}}\rangle ^{2}$	$\langle 33+7{\sqrt {22}}\rangle ^{2}$
13	$\langle 13,2-{\sqrt {-22}}\rangle \langle 13,2+{\sqrt {-22}}\rangle$	$\langle 3-{\sqrt {22}}\rangle \langle 3+{\sqrt {22}}\rangle$
17	Prime	Prime
19	$\langle 19,4-{\sqrt {-22}}\rangle \langle 19,4+{\sqrt {-22}}\rangle$
23	$\langle 1-{\sqrt {-22}}\rangle \langle 1+{\sqrt {-22}}\rangle$
29		$\langle 13-3{\sqrt {22}}\rangle \langle 13+3{\sqrt {22}}\rangle$
31
37
41
43
47

Factorization of 22 in some quadratic integer rings

Of course 22 is composite in all quadratic integer rings. However, in some, one of its two prime factors (2 or 11) is further reducible, and in some, both prime factors are further reducible.

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

Representation of 22 in various bases

Base	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Representation	10110	211	112	42	34	31	26	24	22	20	1A	19	18	17	16	15	14	13	12

Although 22 is a palindromic number in base 10, note that that is the only base that, in the range from 2 to 20, it is palindromic in (it is trivially palindromic in base 21 and all bases higher than 22).

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