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# 23

Please do not rely on any information it contains.

23 is an integer. It is the smallest number of integer-sided boxes that tile a box so that no two boxes share a common length.

## Membership in core sequences

 Odd numbers ..., 17, 19, 21, 23, 25, 27, 29, ... A005408 Prime numbers ..., 13, 17, 19, 23, 29, 31, 37, ... A000040 Squarefree numbers ..., 19, 21, 22, 23, 26, 29, 30, ... A005117 Number of trees with ${\displaystyle n}$ unlabeled nodes ..., 3, 6, 11, 23, 47, 106, 235, ... A000055 Wedderburn-Etherington numbers ..., 3, 6, 11, 23, 46, 98, 207, ... A001190

In Pascal's triangle, 23 occurs twice.

## Sequences pertaining to 23

 Multiples of 23 23, 46, 69, 92, 115, 138, 161, 184, 207, 230, 253, ... A008605 23-gonal numbers 1, 23, 66, 130, 215, 321, 448, 596, 765, 955, 1166, ... A051874 Centered 23-gonal numbers 1, 24, 70, 139, 231, 346, 484, 645, 829, 1036, 1266, ... A069174 Concentric 23-gonal numbers 1, 23, 47, 92, 139, 207, 277, 368, 461, 575, 691, 828, ... A195058 ${\displaystyle 3x+1}$ sequence beginning at 15 15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, ... A033480 ${\displaystyle 3x-1}$ sequence beginning at 83 83, 248, 124, 62, 31, 92, 46, 23, 68, 34, 17, 50, 25, 74, ... A008897 ${\displaystyle 5x+1}$ sequence beginning at 11 ..., 18, 9, 46, 23, 116, 58, 29, 146, 73, 366, 183, 916, ... A259193

## Partitions of 23

There are 1255 partitions of 23. Of these, [FINISH WRITING]

## Roots and powers of 23

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

 ${\displaystyle {\sqrt {23}}}$ 4.79583152 A010479 23 2 529 ${\displaystyle {\sqrt[{3}]{23}}}$ 2.84386697 A010595 23 3 12167 ${\displaystyle {\sqrt[{4}]{23}}}$ 2.18993870 A011019 23 4 279841 ${\displaystyle {\sqrt[{5}]{23}}}$ 1.87217123 A011108 23 5 6436343 ${\displaystyle {\sqrt[{6}]{23}}}$ 1.68637687 23 6 148035889 ${\displaystyle {\sqrt[{7}]{23}}}$ 1.56506560 23 7 3404825447 ${\displaystyle {\sqrt[{8}]{23}}}$ 1.47984414 23 8 78310985281 ${\displaystyle {\sqrt[{9}]{23}}}$ 1.41678220 23 9 1801152661463 ${\displaystyle {\sqrt[{10}]{23}}}$ 1.36827308 23 10 41426511213649 A009967

## Logarithms and 23rd powers

In the OEIS specifically and mathematics in general, ${\displaystyle \log x}$ refers to the natural logarithm of ${\displaystyle x}$, whereas all other bases are specified with a subscript.

If ${\displaystyle n}$ is not a multiple of 47, then either ${\displaystyle n^{23}-1}$ or ${\displaystyle n^{23}+1}$ is. Hence the formula for the Legendre symbol ${\displaystyle \left({\frac {a}{47}}\right)=a^{23}{\bmod {4}}7}$.

As above, irrational numbers in the following table are truncated to eight decimal places.

 ${\displaystyle \log _{23}2}$ 0.221065 A152882 ${\displaystyle \log _{2}23}$ 4.52356 A155793 2 23 8388608 ${\displaystyle \log _{23}e}$ 0.318929 ${\displaystyle \log 23}$ 3.13549 A016646 ${\displaystyle e^{23}}$ 9.74480344... × 10 9 ${\displaystyle \log _{23}3}$ 0.350379 A153099 ${\displaystyle \log _{3}23}$ 2.85405 A155808 3 23 94143178827 ${\displaystyle \log _{23}\pi }$ 0.365088 ${\displaystyle \log _{\pi }23}$ 2.73907 ${\displaystyle \pi ^{23}}$ 2.71923706 × 10 11 ${\displaystyle \log _{23}4}$ 0.442129 A153163 ${\displaystyle \log _{4}23}$ 2.26178 A155818 4 23 70368744177664 ${\displaystyle \log _{23}5}$ 0.513296 A153457 ${\displaystyle \log _{5}23}$ 1.94819 A155821 5 23 11920928955078125 ${\displaystyle \log _{23}6}$ 0.571444 A153613 ${\displaystyle \log _{6}23}$ 1.74995 A155823 6 23 789730223053602816 ${\displaystyle \log _{23}7}$ 0.620607 A153735 ${\displaystyle \log _{7}23}$ 1.61133 A155824 7 23 27368747340080916343 ${\displaystyle \log _{23}8}$ 0.663194 A154006 ${\displaystyle \log _{8}23}$ 1.50785 A155827 8 23 590295810358705651712 ${\displaystyle \log _{23}9}$ 0.700759 A154102 ${\displaystyle \log _{9}23}$ 1.42702 A155829 9 23 8862938119652501095929 ${\displaystyle \log _{23}10}$ 0.734361 A154173 ${\displaystyle \log _{10}23}$ 1.36173 A155830 10 23 100000000000000000000000

(See A010811 for the 23rd powers of integers).

## Values for number theoretic functions with 23 as an argument

 ${\displaystyle \mu (23)}$ –1 ${\displaystyle M(23)}$ –4 ${\displaystyle \pi (23)}$ 9 ${\displaystyle \sigma _{1}(23)}$ 24 ${\displaystyle \sigma _{0}(23)}$ 1 ${\displaystyle \phi (23)}$ 22 ${\displaystyle \Omega (23)}$ 1 ${\displaystyle \omega (23)}$ 1 ${\displaystyle \lambda (23)}$ 22 This is the Carmichael lambda function. ${\displaystyle \lambda (23)}$ 1 This is the Liouville lambda function. ${\displaystyle \zeta (23)}$ 1.0000001192199... 22! 25852016738884976640000 ${\displaystyle \Gamma (23)}$ 1124000727777607680000

## Factorization of some small integers in a quadratic integer ring adjoining the square roots of −23, 23

The commutative quadratic integer ring with unity ${\displaystyle \scriptstyle \mathbb {Z} [{\sqrt {23}}]\,}$, with units of the form ${\displaystyle \scriptstyle \pm (24+5{\sqrt {23}})^{n}\,}$ (${\displaystyle \scriptstyle n\,\in \,\mathbb {Z} \,}$), is a unique factorization domain. ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-23}})}}$, on the other hand, is not only not a UFD, it has class number 3, with the consequence that for some numbers with more than one distinct factorization, one factorization might have more prime factors than another.

 ${\displaystyle n}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-23}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {23}}]}$ 2 Irreducible ${\displaystyle (5-{\sqrt {23}})(5+{\sqrt {23}})}$ 3 Prime 4 2 2 ${\displaystyle (5-{\sqrt {23}})^{2}(5+{\sqrt {23}})^{2}}$ 5 Prime 6 2 × 3 OR ${\displaystyle \left({\frac {1}{2}}-{\frac {\sqrt {-23}}{2}}\right)\left({\frac {1}{2}}+{\frac {\sqrt {-23}}{2}}\right)}$ ${\displaystyle (5-{\sqrt {23}})(5+{\sqrt {23}})3}$ 7 Prime ${\displaystyle (-1)(4-{\sqrt {23}})(4+{\sqrt {23}})}$ 8 2 3 OR ${\displaystyle \left({\frac {3}{2}}-{\frac {\sqrt {-23}}{2}}\right)\left({\frac {3}{2}}+{\frac {\sqrt {-23}}{2}}\right)}$ ${\displaystyle (5-{\sqrt {23}})^{3}(5+{\sqrt {23}})^{3}}$ 9 3 2 10 2 × 5 ${\displaystyle (5-{\sqrt {23}})(5+{\sqrt {23}})5}$ 11 Prime ${\displaystyle (-1)(9-2{\sqrt {23}})(9+2{\sqrt {23}})}$ 12 2 2 × 3 OR ${\displaystyle \left({\frac {5}{2}}-{\frac {\sqrt {-23}}{2}}\right)\left({\frac {5}{2}}+{\frac {\sqrt {-23}}{2}}\right)}$ ${\displaystyle (5-{\sqrt {23}})^{2}(5+{\sqrt {23}})^{2}3}$ 13 Irreducible ${\displaystyle (6-{\sqrt {23}})(6+{\sqrt {23}})}$ 14 2 × 7 ${\displaystyle (-1)(5-{\sqrt {23}})(5+{\sqrt {23}})(4-{\sqrt {23}})(4+{\sqrt {23}})}$ 15 3 × 5 16 2 4 ${\displaystyle (5-{\sqrt {23}})^{4}(5+{\sqrt {23}})^{4}}$ 17 Irreducible Prime 18 2 × 3 2 OR ${\displaystyle \left({\frac {7}{2}}-{\frac {\sqrt {-23}}{2}}\right)\left({\frac {7}{2}}+{\frac {\sqrt {-23}}{2}}\right)}$ ${\displaystyle (5-{\sqrt {23}})(5+{\sqrt {23}})3^{2}}$ 19 Irreducible ${\displaystyle (-1)(2-{\sqrt {23}})(2+{\sqrt {23}})}$ 20 2 2 × 3 ${\displaystyle (5-{\sqrt {23}})(5+{\sqrt {23}})5}$

Perhaps it does not need to be said that ${\displaystyle (-1)(1-{\sqrt {23}})(1+{\sqrt {23}})}$ is not a distinct factorization of 22, since this is a UFD and we readily see that ${\displaystyle (-5-{\sqrt {23}})(9-2{\sqrt {23}})=1+{\sqrt {23}}}$.

Ideals really help us make sense of multiple distinct factorizations in ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-23}})}}$.

 ${\displaystyle p}$ Factorization of ${\displaystyle \langle p\rangle }$ In ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-23}})}}$ In ${\displaystyle \mathbb {Z} [{\sqrt {23}}]}$ 2 ${\displaystyle \left\langle 2,{\frac {1}{2}}-{\frac {\sqrt {-23}}{2}}\right\rangle \left\langle 2,{\frac {1}{2}}+{\frac {\sqrt {-23}}{2}}\right\rangle }$ ${\displaystyle \langle 5+{\sqrt {23}}\rangle ^{2}}$ 3 ${\displaystyle \langle 3,1-{\sqrt {-23}}\rangle \langle 3,1+{\sqrt {-23}}\rangle }$ Prime 5 Prime 7 Prime ${\displaystyle \langle 4-{\sqrt {23}}\rangle \langle 4+{\sqrt {23}}\rangle }$ 11 Prime ${\displaystyle \langle 9-{\sqrt {23}}\rangle \langle 9+{\sqrt {23}}\rangle }$ 13 ${\displaystyle \langle 13,4-{\sqrt {-23}}\rangle \langle 13,4+{\sqrt {-23}}\rangle }$ ${\displaystyle \langle 6-{\sqrt {23}}\rangle \langle 6+{\sqrt {23}}\rangle }$ 17 Prime 19 Prime ${\displaystyle \langle 2-{\sqrt {23}}\rangle \langle 2+{\sqrt {23}}\rangle }$ 23 ${\displaystyle \langle {\sqrt {-23}}\rangle ^{2}}$ ${\displaystyle \langle {\sqrt {23}}\rangle ^{2}}$ 29 ${\displaystyle \langle 29,8-{\sqrt {-23}}\rangle \langle 29,8+{\sqrt {-23}}\rangle }$ ${\displaystyle \langle 11-2{\sqrt {23}}\rangle \langle 11+2{\sqrt {23}}\rangle }$ 31 37 41 43 47

PLACEHOLDER

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## Representation of 23 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 10111 212 113 43 35 32 27 25 23 21 1B 1A 19 16 17 16 15 14 13

 ${\displaystyle -1}$ 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