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# 133

Please do not rely on any information it contains.

133 is an integer. The proper divisors of 133 are 1, 7, 19, which add up to 27, and that is a divisor of ${\displaystyle \phi (133)=108}$. No smaller number has this property.

## Membership in core sequences

 Odd numbers ..., 127, 129, 131, 133, 135, 137, 139, ... A005408 Composite numbers ..., 129, 130, 132, 133, 134, 135, 136, ... A002808 Semiprimes ..., 122, 123, 129, 133, 134, 141, 142, ... A001358 Squarefree numbers ..., 129, 130, 131, 133, 134, 137, 138, ... A005117 Lucky numbers ..., 115, 127, 129, 133, 135, 141, 151, ... A000959 Loeschian numbers ..., 124, 127, 129, 133, 139, 144, 147, ... A003136

## Sequences pertaining to 133

 Multiples of 133 0, 133, 266, 399, 532, 665, 798, 931, 1064, 1197, 1330, ... ${\displaystyle 3x-1}$ sequence beginning at 89 89, 266, 133, 398, 199, 596, 298, 149, 446, 223, 668, 334, ... A008900

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## Roots and powers of 133

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

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## Factorization of some small integers in a quadratic integer ring adjoining ${\displaystyle {\sqrt {-133}}}$, ${\displaystyle {\sqrt {133}}}$

${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {133}})}}$ is a unique factorization domain. Units in ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {133}})}}$ are of the form ${\displaystyle \left({\frac {173}{2}}+{\frac {15{\sqrt {133}}}{2}}\right)^{n}}$.

 ${\displaystyle n}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {133}})}}$ 2 Prime 3 ${\displaystyle (-1)\left({\frac {11}{2}}-{\frac {\sqrt {133}}{2}}\right)\left({\frac {11}{2}}+{\frac {\sqrt {133}}{2}}\right)}$ 4 2 2 5 Prime 6 ${\displaystyle (-1)2\left({\frac {11}{2}}\pm {\frac {\sqrt {133}}{2}}\right)}$ 7 ${\displaystyle \left({\frac {35}{2}}-{\frac {3{\sqrt {133}}}{2}}\right)\left({\frac {35}{2}}+{\frac {3{\sqrt {133}}}{2}}\right)}$ 8 2 3 9 ${\displaystyle \left({\frac {11}{2}}\pm {\frac {\sqrt {133}}{2}}\right)^{2}}$ 10 2 × 5 11 ${\displaystyle \left({\frac {81}{2}}-{\frac {7{\sqrt {133}}}{2}}\right)\left({\frac {81}{2}}+{\frac {7{\sqrt {133}}}{2}}\right)}$ 12 2 2 × 3 13 ${\displaystyle (-1)\left({\frac {9}{2}}-{\frac {\sqrt {133}}{2}}\right)\left({\frac {9}{2}}+{\frac {\sqrt {133}}{2}}\right)}$ 14 ${\displaystyle 2\left({\frac {35}{2}}\pm {\frac {3{\sqrt {133}}}{2}}\right)}$ 15 ${\displaystyle (-1)\left({\frac {11}{2}}\pm {\frac {\sqrt {133}}{2}}\right)5}$ 16 2 4 17 Prime 18 ${\displaystyle 2\left({\frac {11}{2}}\pm {\frac {\sqrt {133}}{2}}\right)^{2}}$ 19 ${\displaystyle (-1)\left({\frac {57}{2}}-{\frac {5{\sqrt {133}}}{2}}\right)\left({\frac {57}{2}}+{\frac {5{\sqrt {133}}}{2}}\right)}$ 20 2 2 × 5 21 ${\displaystyle (-1)\left({\frac {11}{2}}\pm {\frac {\sqrt {133}}{2}}\right)\left({\frac {35}{2}}\pm {\frac {3{\sqrt {133}}}{2}}\right)}$

Unlike ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {133}})}}$, ${\displaystyle \mathbb {Z} [{\sqrt {-133}}]}$ is not a unique factorization, and what's more, it has class number 4. Here we will give a few examples of numbers with more than one distinct factorization in ${\displaystyle \mathbb {Z} [{\sqrt {-133}}]}$ in which the factorizations have differing numbers of irreducible factors.

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TABLE GOES HERE

## Representation of 133 in various bases

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REMARKS GO HERE

 ${\displaystyle -1}$ 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