133

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133 is an integer. The proper divisors of 133 are 1, 7, 19, which add up to 27, and that is a divisor of $\phi (133)=108$. 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, ...
$3x-1$ sequence beginning at 89	89, 266, 133, 398, 199, 596, 298, 149, 446, 223, 668, 334, ...	A008900

Partitions of 133

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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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Logarithms and 133rd powers

REMARKS

TABLE

Values for number theoretic functions with 133 as an argument

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

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

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

Unlike ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {133}})}$, $\mathbb {Z} [{\sqrt {-133}}]$ 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 $\mathbb {Z} [{\sqrt {-133}}]$ in which the factorizations have differing numbers of irreducible factors.

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Factorization of 133 in some quadratic integer rings

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TABLE GOES HERE

Representation of 133 in various bases

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REMARKS GO HERE

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