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# 22

Please do not rely on any information it contains.

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 ${\displaystyle 3x+1}$ 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.

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

 ${\displaystyle \mu (22)}$ 1 ${\displaystyle M(22)}$ –3 ${\displaystyle \pi (22)}$ 8 ${\displaystyle \sigma _{1}(22)}$ 36 ${\displaystyle \sigma _{0}(22)}$ 4 ${\displaystyle \phi (22)}$ 10 ${\displaystyle \Omega (22)}$ 2 ${\displaystyle \omega (22)}$ 2 ${\displaystyle \lambda (22)}$ 10 This is the Carmichael lambda function. ${\displaystyle \lambda (22)}$ 1 This is the Liouville lambda function. ${\displaystyle \zeta (22)}$ 1.00000023845... (see A013668). 22! 1124000727777607680000 ${\displaystyle \Gamma (22)}$ 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 ${\displaystyle \scriptstyle \mathbb {Z} [{\sqrt {22}}]\,}$, with units of the form ${\displaystyle \scriptstyle \pm (197+42{\sqrt {22}})^{n}\,}$ (${\displaystyle \scriptstyle n\,\in \,\mathbb {Z} \,}$), is a unique factorization domain.

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

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

Ideals help us make sense of multiple distinct factorizations.

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

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

## 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).

 ${\displaystyle -1}$ 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