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# 38

Please do not rely on any information it contains.

38 is an integer.

## Membership in core sequences

 Even numbers ..., 32, 34, 36, 38, 40, 42, 44, ... A005843 Squarefree numbers ..., 34, 35, 37, 38, 39, 41, 42, ... A005117

## Sequences pertaining to 38

 Multiples of 38 0, 38, 76, 114, 152, 190, 228, 266, 304, 342, 380, 418, 456, ... ${\displaystyle 3x+1}$ sequence starting at 33 33, 100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, ... A008880

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## Roots and powers of 38

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

 ${\displaystyle {\sqrt {38}}}$ 6.16441400 A010492 38 2 1444 ${\displaystyle {\sqrt[{3}]{38}}}$ 3.36197540 A010609 38 3 54872 ${\displaystyle {\sqrt[{4}]{38}}}$ 2.48282379 A011032 38 4 2085136 ${\displaystyle {\sqrt[{5}]{38}}}$ 2.06993505 A011123 38 5 79235168 ${\displaystyle {\sqrt[{6}]{38}}}$ 1.83356903 38 6 3010936384 ${\displaystyle {\sqrt[{7}]{38}}}$ 1.68144772 38 7 114415582592 ${\displaystyle {\sqrt[{8}]{38}}}$ 1.57569787 38 8 4347792138496 ${\displaystyle {\sqrt[{9}]{38}}}$ 1.49806794 38 9 165216101262848 ${\displaystyle {\sqrt[{10}]{38}}}$ 1.43872688 38 10 6278211847988224 A009982

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## Values for number theoretic functions with 38 as an argument

 ${\displaystyle \mu (38)}$ 1 ${\displaystyle M(38)}$ −1 ${\displaystyle \pi (38)}$ 11 ${\displaystyle \sigma _{1}(38)}$ 60 ${\displaystyle \sigma _{0}(38)}$ 4 ${\displaystyle \phi (38)}$ 18 ${\displaystyle \Omega (38)}$ 2 ${\displaystyle \omega (38)}$ 2 ${\displaystyle \lambda (38)}$ 18 This is the Carmichael lambda function. ${\displaystyle \lambda (38)}$ 1 This is the Liouville lambda function. ${\displaystyle \zeta (38)}$ 38! 523022617466601111760007224100074291200000000 ${\displaystyle \Gamma (38)}$ 13763753091226345046315979581580902400000000

## Factorization of some small integers in a quadratic integer ring adjoining the square roots of −38, 38

The commutative quadratic integer ring with unity ${\displaystyle \mathbb {Z} [{\sqrt {38}}]}$, with units of the form ${\displaystyle \pm (37+6{\sqrt {38}})^{n}\,}$ (${\displaystyle n\in \mathbb {Z} }$), is a unique factorization domain.

 ${\displaystyle n}$ ${\displaystyle \mathbb {Z} [{\sqrt {38}}]}$ 2 ${\displaystyle (-1)(6-{\sqrt {38}})(6+{\sqrt {38}})}$ 3 Prime 4 ${\displaystyle (6-{\sqrt {38}})^{2}(6+{\sqrt {38}})^{2}}$ 5 Prime 6 ${\displaystyle (-1)(6-{\sqrt {38}})(6+{\sqrt {38}})3}$ 7 Prime 8 ${\displaystyle (-1)(6-{\sqrt {38}})^{3}(6+{\sqrt {38}})^{3}}$ 9 3 2 10 ${\displaystyle (-1)(6-{\sqrt {38}})(6+{\sqrt {38}})5}$ 11 ${\displaystyle (7-{\sqrt {38}})(7+{\sqrt {38}})}$ 12 ${\displaystyle (6-{\sqrt {38}})^{2}(6+{\sqrt {38}})^{2}3}$ 13 ${\displaystyle (-1)(5-{\sqrt {38}})(5+{\sqrt {38}})}$ 14 ${\displaystyle (-1)(6-{\sqrt {38}})(6+{\sqrt {38}})7}$ 15 3 × 5 16 ${\displaystyle (6-{\sqrt {38}})^{4}(6+{\sqrt {38}})^{4}}$ 17 ${\displaystyle (13-2{\sqrt {38}})(13+2{\sqrt {38}})}$ 18 ${\displaystyle (-1)(6-{\sqrt {38}})(6+{\sqrt {38}})3^{2}}$ 19 ${\displaystyle (19-3{\sqrt {38}})(19+3{\sqrt {38}})}$ 20 ${\displaystyle (6-{\sqrt {38}})^{2}(6+{\sqrt {38}})^{2}5}$

Unlike ${\displaystyle \mathbb {Z} [{\sqrt {38}}]}$, ${\displaystyle \mathbb {Z} [{\sqrt {-38}}]}$ is not a unique factorization domain at all, having class number 6. But the window of 2 through 21 does not provide as interesting a window for the of the [FINISH WRITING]

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## Representation of 38 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 100110 1102 212 123 102 53 46 42 38 35 32 2C 2A 28 26 24 22 20 1I

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 ${\displaystyle -1}$ 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