23

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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 $n$ 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
$3x+1$ sequence beginning at 15	15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, ...	A033480
$3x-1$ sequence beginning at 83	83, 248, 124, 62, 31, 92, 46, 23, 68, 34, 17, 50, 25, 74, ...	A008897
$5x+1$ 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.

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

Logarithms and 23rd powers

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

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

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

$\log _{23}2$	0.22106472	A152882	$\log _{2}23$	4.52356195	A155793	2 23	8388608
$\log _{23}e$	0.31892898		$\log 23$	3.13549421	A016646	$e^{23}$	9.74480344... × 10 9
$\log _{23}3$	0.35037930	A153099	$\log _{3}23$	2.85404983	A155808	3 23	94143178827
$\log _{23}\pi$	0.36508754		$\log _{\pi }23$	2.73906906		$\pi ^{23}$	2.71923706 × 10 11
$\log _{23}4$	0.44212945	A153163	$\log _{4}23$	2.26178097	A155818	4 23	70368744177664
$\log _{23}5$	0.51329640	A153457	$\log _{5}23$	1.94819209	A155821	5 23	11920928955078125
$\log _{23}6$	0.57144403	A153613	$\log _{6}23$	1.74995264	A155823	6 23	789730223053602816
$\log _{23}7$	0.62060715	A153735	$\log _{7}23$	1.61132528	A155824	7 23	27368747340080916343
$\log _{23}8$	0.66319418	A154006	$\log _{8}23$	1.50785398	A155827	8 23	590295810358705651712
$\log _{23}9$	0.70075861	A154102	$\log _{9}23$	1.42702491	A155829	9 23	8862938119652501095929
$\log _{23}10$	0.73436113	A154173	$\log _{10}23$	1.36172783	A155830	10 23	100000000000000000000000

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

Values for number theoretic functions with 23 as an argument

$\mu (23)$	–1
$M(23)$	–4
$\pi (23)$	9
$\sigma _{1}(23)$	24
$\sigma _{0}(23)$	1
$\phi (23)$	22
$\Omega (23)$	1
$\omega (23)$	1
$\lambda (23)$	22	This is the Carmichael lambda function.
$\lambda (23)$	1	This is the Liouville lambda function.
$\zeta (23)$	1.0000001192199...
22!	25852016738884976640000
$\Gamma (23)$	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 $\scriptstyle \mathbb {Z} [{\sqrt {23}}]\,$, with units of the form $\scriptstyle \pm (24+5{\sqrt {23}})^{n}\,$ ($\scriptstyle n\,\in \,\mathbb {Z} \,$), is a unique factorization domain. ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-23}})}$, 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.

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

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

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

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

Factorization of 23 in some quadratic integer rings

PLACEHOLDER

TABLE GOES HERE

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

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