21

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21 is an integer, the smallest number of distinct squares needed to tile a square (see A006983).

Membership in core sequences

Odd numbers	..., 15, 17, 19, 21, 23, 25, 27, ...	A005408
Composite numbers	..., 16, 18, 20, 21, 22, 24, 25, ...	A002808
Semiprimes	..., 10, 14 15, 21, 22, 25, 26, ...	A001358
Squarefree numbers	..., 15, 17, 19, 21, 22, 23, 26, ...	A005117
Fibonacci numbers	..., 5, 8, 13, 21, 34, 55, 89, ...	A000045
Triangular numbers	..., 6, 10, 15, 21, 28, 36, 45, ...	A000217

Sequences pertaining to 21

Multiples of 21	0, 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, ...	A008603
21-gonal numbers	1, 21, 60, 118, 195, 291, 406, 540, 693, 865, 1056, ...	A051873
Centered 21-gonal numbers	1, 22, 64, 127, 211, 316, 442, 589, 757, 946, 1156, ...	A069178
Concentric 21-gonal numbers	1, 21, 43, 84, 127, 189, 253, 336, 421, 525, 631, ...	A195049
$3x+1$ sequence beginning at 21	21, 64, 32, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, ...	A033481
$3x-1$ sequence beginning at 84	84, 42, 21, 62, 31, 92, 46, 23, 68, 34, 17, 50, 25, 74, ...	A008898
$5x-1$ sequence beginning at 50	..., 6, 3, 14, 7, 34, 17, 84, 42, 21, 104, 52, 26, 13, 64, ...	A090691

Partitions of 21

There are 792 partitions of 21.

The Goldbach representations of 21 are: 2 + 19 = 3 + 5 + 13 = 3 + 7 + 11 = 5 + 5 + 11 = 7 + 7 + 7 = 21.

Roots and powers of 21

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

${\sqrt {21}}$	4.58257569	A010477	21 2	441
${\sqrt[{3}]{21}}$	2.75892417	A010593	21 3	9261
${\sqrt[{4}]{21}}$	2.14069514	A011017	21 4	194481
${\sqrt[{5}]{21}}$	1.83841628	A011106	21 5	4084101
${\sqrt[{6}]{21}}$	1.66100095		21 6	85766121
${\sqrt[{7}]{21}}$	1.54485766		21 7	1801088541
${\sqrt[{8}]{21}}$	1.46311145		21 8	37822859361
${\sqrt[{9}]{21}}$	1.40253353		21 9	794280046581
${\sqrt[{10}]{21}}$	1.35588210		21 10	16679880978201
				A009965

Logarithms and 21st 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.

From the basic properties of exponentiation, it follows that all 21st powers are seventh powers of cubes, as well as cubes of seventh powers.

If $n$ is not a multiple of 43, then either $n^{21}-1$ or $n^{21}+1$ is. Hence the formula for the Legendre symbol $\left({\frac {a}{43}}\right)=a^{21}\mod 43$.

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

$\log _{21}2$	0.22767024	A152825	$\log _{2}21$	4.39231742	A155536	2 21	2097152
$\log _{21}e$	0.32845873		$\log 21$	3.04452243	A016644	$e^{21}$
$\log _{21}3$	0.36084880	A153097	$\log _{3}21$	2.77124374	A155541	3 21	10460353203
$\log _{21}\pi$	0.37599653		$\log _{\pi }21$	2.65959898		$\pi ^{21}$
$\log _{21}4$	0.45534049	A153131	$\log _{4}21$	2.19615871	A155545	4 21	4398046511104
$\log _{21}5$	0.52863394	A153455	$\log _{5}21$	1.89166814	A155553	5 21	476837158203125
$\log _{21}6$	0.58851905	A153611	$\log _{6}21$	1.69918032	A155554	6 21	21936950640377856
$\log _{21}7$	0.63915119	A153632	$\log _{7}21$	1.56457503	A155591	7 21	558545864083284007
$\log _{21}8$	0.68301074	A153895	$\log _{8}21$	1.46410580	A155675	8 21	9223372036854775808
$\log _{21}9$	0.72169761	A154020	$\log _{9}21$	1.38562187	A155676	9 21	109418989131512359209
$\log _{21}10$	0.75630419	A154171	$\log _{10}21$	1.32221929	A155677	10 21	1000000000000000000000
$\log _{21}11$	0.78760965	A154192	$\log _{11}21$	1.26966447	A155678	11 21	7400249944258160101211
$\log _{21}12$	0.81618930	A154213	$\log _{12}21$	1.22520596	A155679	12 21	46005119909369701466112

(See A010809 for the 21st powers of integers).

Values for number theoretic functions with 21 as an argument

$\mu (21)$	1
$M(21)$	–2
$\pi (21)$	8
$\sigma _{1}(21)$	32
$\sigma _{0}(21)$	4
$\phi (21)$	12
$\Omega (21)$	2
$\omega (21)$	2
$\lambda (21)$	6	This is the Carmichael lambda function.
$\lambda (21)$	1	This is the Liouville lambda function.
$\zeta (21)$	1.000000476932986787806463116719604373...
21!	51090942171709440000
$\Gamma (21)$	2432902008176640000

Factorization of some small integers in a quadratic integer ring adjoining the square roots of −21, 21

${\mathcal {O}}_{\mathbb {Q} ({\sqrt {21}})}$ is a unique factorization domain, but $\mathbb {Z} [{\sqrt {-21}}]$ is not. The fundamental unit in ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {21}})}$ is $\left({\frac {5}{2}}+{\frac {\sqrt {21}}{2}}\right)$, with norm 1.

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

It is worth emphasizing that $({\sqrt {21}})^{2}$ is not a distinct factorization of 21. Observe that $\left({\frac {3}{2}}+{\frac {\sqrt {21}}{2}}\right)\left({\frac {7}{2}}-{\frac {\sqrt {21}}{2}}\right)={\sqrt {21}}$.

Ideals really help us make sense of multiple distinct factorizations in $\mathbb {Z} [{\sqrt {-21}}]$.

$p$	Factorization of $\langle p\rangle$
	In $\mathbb {Z} [{\sqrt {-21}}]$	In ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {21}})}$
2	$\langle 2,{\sqrt {-21}}\rangle ^{2}$	Prime
3	$\langle 3,{\sqrt {-21}}\rangle ^{2}$	$\left\langle {\frac {3}{2}}+{\frac {\sqrt {21}}{2}}\right\rangle ^{2}$
5	$\langle 5,2-{\sqrt {-21}}\rangle \langle 5,2+{\sqrt {-21}}\rangle$
7	$\langle 7,{\sqrt {-21}}\rangle ^{2}$	$\left\langle {\frac {7}{2}}+{\frac {\sqrt {21}}{2}}\right\rangle ^{2}$
11	$\langle 11,1-{\sqrt {-21}}\rangle \langle 11,1+{\sqrt {-21}}\rangle$	Prime
13	Prime
17	$\langle 17,8-{\sqrt {-21}}\rangle \langle 17,8+{\sqrt {-21}}\rangle$	$\langle 2-{\sqrt {21}}\rangle \langle 2+{\sqrt {21}}\rangle$
19	$\langle 19,6-{\sqrt {-21}}\rangle \langle 19,6+{\sqrt {-21}}\rangle$	Prime
23	$\langle 23,5-{\sqrt {-21}}\rangle \langle 23,5+{\sqrt {-21}}\rangle$
29
31
37
41
43
47

Factorization of 21 in some quadratic integer rings

Since 21 is composite in $\mathbb {Z}$, being the product of 3 and 7, it follows that it is also composite in all quadratic integer rings. However, in some rings it can be factorized further, and in some rings it has more than one factorization.

$\mathbb {Z} [i]$	3 × 7
$\mathbb {Z} [{\sqrt {-2}}]$	$(1-{\sqrt {-2}})(1+{\sqrt {-2}})7$	$\mathbb {Z} [{\sqrt {2}}]$	$3(3-{\sqrt {2}})(3+{\sqrt {2}})$
$\mathbb {Z} [\omega ]$	$(-1)(1+2\omega )^{2}(-2+\omega )(-2+\omega ^{2})$	$\mathbb {Z} [{\sqrt {3}}]$	$({\sqrt {3}})^{2}7$
$\mathbb {Z} [{\sqrt {-5}}]$	3 × 7 OR $(4-{\sqrt {-5}})(4+{\sqrt {-5}})$ OR $(1-2{\sqrt {-5}})(1+2{\sqrt {-5}})$	$\mathbb {Z} [\phi ]$	3 × 7
$\mathbb {Z} [{\sqrt {-6}}]$	$3(1-{\sqrt {-6}})(1+{\sqrt {-6}})$	$\mathbb {Z} [{\sqrt {6}}]$	$(3-{\sqrt {6}})(3+{\sqrt {6}})7$
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}$	$(-1)3({\sqrt {-7}})^{2}$	$\mathbb {Z} [{\sqrt {7}}]$	$(-1)(2-{\sqrt {7}})(2+{\sqrt {7}})({\sqrt {7}})^{2}$
$\mathbb {Z} [{\sqrt {-10}}]$	3 × 7	$\mathbb {Z} [{\sqrt {10}}]$	3 × 7
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}$	$\left({\frac {1}{2}}-{\frac {\sqrt {-11}}{2}}\right)\left({\frac {1}{2}}+{\frac {\sqrt {-11}}{2}}\right)7$	$\mathbb {Z} [{\sqrt {11}}]$	$(-1)3(2-{\sqrt {11}})(2+{\sqrt {11}})$
$\mathbb {Z} [{\sqrt {-13}}]$	3 × 7	${\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}$	$(-1)\left({\frac {1}{2}}-{\frac {\sqrt {13}}{2}}\right)\left({\frac {1}{2}}+{\frac {\sqrt {13}}{2}}\right)7$
$\mathbb {Z} [{\sqrt {-14}}]$		$\mathbb {Z} [{\sqrt {14}}]$	$(-1)3(7-2{\sqrt {14}})(7+2{\sqrt {14}})$
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}$		$\mathbb {Z} [{\sqrt {15}}]$	3 × 7 OR $(6-{\sqrt {15}})(6+{\sqrt {15}})$
$\mathbb {Z} [{\sqrt {-17}}]$	3 × 7 OR $(2-{\sqrt {-17}})(2+{\sqrt {-17}})$	${\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}$	3 × 7
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}$	$3\left({\frac {3}{2}}-{\frac {\sqrt {-19}}{2}}\right)\left({\frac {3}{2}}+{\frac {\sqrt {-19}}{2}}\right)$	$\mathbb {Z} [{\sqrt {19}}]$	$(-1)(4-{\sqrt {19}})(4+{\sqrt {19}})7$

Of particular interest, note how 21 has three distinct factorizations in $\mathbb {Z} [{\sqrt {-5}}]$.

Representation of 21 in various bases

Base	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Representation	10101	210	111	41	33	30	25	23	21	1A	19	18	17	16	15	14	13	12	11

Notice how 21 is a Harshad number in quite a few different bases: 2, 3, 4, 7, 8, 10, 15, 19. Also, it is palindromic in bases 2, 4, 6.

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