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# 163

Please do not rely on any information it contains.

163 is an integer.

## Membership in core sequences

 Odd numbers ..., 157, 159, 161, 163, 165, 167, 169, ... A005408 Prime numbers ..., 149, 151, 157, 163, 167, 173, 179, ... A000040 Lucky numbers ..., 141, 151, 159, 163, 169, 171, 189, ... A001358 Squarefree numbers ..., 158, 159, 161, 163, 165, 166, 167, ... A005117 Loeschian numbers ..., 151, 156, 157, 163, 169, 171, 172, ... A003136

In Pascal's triangle, 163 occurs twice.

## Sequences pertaining to 163

 Multiples of 163 0, 163, 326, 489, 652, 815, 978, 1141, 1304, ... ${\displaystyle 3x+1}$ sequence beginning at 163 163, 490, 245, 736, 368, 184, 92, 46, 23, 70, ...

PLACEHOLDER

## Roots and powers of 163

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

 ${\displaystyle {\sqrt {163}}}$ 12.76714533 A210963 1632 26569 ${\displaystyle {\sqrt[{3}]{163}}}$ 5.46255557 1633 4330747 ${\displaystyle {\sqrt[{4}]{163}}}$ 3.57311423 1634 705911761 ${\displaystyle {\sqrt[{5}]{163}}}$ 2.76973054 1635 115063617043 ${\displaystyle {\sqrt[{6}]{163}}}$ 2.33721106 1636 18755369578009 ${\displaystyle {\sqrt[{7}]{163}}}$ 2.07026910 1637 3057125241215467 ${\displaystyle {\sqrt[{8}]{163}}}$ 1.89026829 1638 498311414318121121 ${\displaystyle {\sqrt[{9}]{163}}}$ 1.76115922 1639 81224760533853742723 ${\displaystyle {\sqrt[{10}]{163}}}$ 1.66425074 16310 13239635967018160063849

REMARKS

TABLE

## Values for number theoretic functions with 163 as an argument

 ${\displaystyle \mu (163)}$ −1 ${\displaystyle M(163)}$ 0 ${\displaystyle \pi (163)}$ 38 ${\displaystyle \sigma _{1}(163)}$ 164 ${\displaystyle \sigma _{0}(163)}$ 2 ${\displaystyle \phi (163)}$ 162 ${\displaystyle \Omega (163)}$ 1 ${\displaystyle \omega (163)}$ 1 ${\displaystyle \lambda (163)}$ 162 This is the Carmichael lambda function. ${\displaystyle \lambda (163)}$ −1 This is the Liouville lambda function.

PLACEHOLDER

## Factorization of 163 in some quadratic integer rings

As was mentioned above, 163 is prime in ${\displaystyle \mathbb {Z} }$. But it has different factorizations in some quadratic integer rings.

 ${\displaystyle \mathbb {Z} [i]}$ Prime ${\displaystyle \mathbb {Z} [{\sqrt {-2}}]}$ ${\displaystyle (1-9{\sqrt {-2}})(1+9{\sqrt {-2}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {2}}]}$ Prime ${\displaystyle \mathbb {Z} [\omega ]}$ ${\displaystyle (3-11\omega )(14+11\omega )}$ ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {-5}}]}$ Irreducible ${\displaystyle \mathbb {Z} [\phi ]}$ ${\displaystyle \mathbb {Z} [{\sqrt {-6}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {6}}]}$ ${\displaystyle (53-21{\sqrt {6}})(53+21{\sqrt {6}})}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}}$ ${\displaystyle (10-3{\sqrt {-7}})(10+3{\sqrt {-7}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {7}}]}$ Prime ${\displaystyle \mathbb {Z} [{\sqrt {-10}}]}$ Irreducible ${\displaystyle \mathbb {Z} [{\sqrt {10}}]}$ Irreducible ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}}$ ${\displaystyle (8-3{\sqrt {-11}})(8+3{\sqrt {-11}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {11}}]}$ Prime ${\displaystyle \mathbb {Z} [{\sqrt {-13}}]}$ Irreducible ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {14}}]}$ ${\displaystyle (87-23{\sqrt {14}})(87+23{\sqrt {14}})}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {15}}]}$ Irreducible ${\displaystyle \mathbb {Z} [{\sqrt {-17}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}}$ Prime ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}}$ ${\displaystyle (12-{\sqrt {-19}})(12+{\sqrt {-19}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {19}}]}$

## Representation of 163 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 10100011 20001 2203 1123 431 322 243 201 163 139 117 C7 B9 AD 1A 9A 91 8B 83

 ${\displaystyle -i}$ ${\displaystyle i}$ ${\displaystyle -1}$ 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 1729