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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, ... $3x+1$ 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.

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

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TABLE

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

 $\mu (38)$ 1 $M(38)$ −1 $\pi (38)$ 11 $\sigma _{1}(38)$ 60 $\sigma _{0}(38)$ 4 $\phi (38)$ 18 $\Omega (38)$ 2 $\omega (38)$ 2 $\lambda (38)$ 18 This is the Carmichael lambda function. $\lambda (38)$ 1 This is the Liouville lambda function. $\zeta (38)$ 38! 523022617466601111760007224100074291200000000 $\Gamma (38)$ 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 $\mathbb {Z} [{\sqrt {38}}]$ , with units of the form $\pm (37+6{\sqrt {38}})^{n}\,$ ($n\in \mathbb {Z}$ ), is a unique factorization domain.

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

PLACEHOLDER

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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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 $-1$ 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