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21

Please do not rely on any information it contains.

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 ${\displaystyle 3x+1}$ sequence beginning at 21 21, 64, 32, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, ... A033481 ${\displaystyle 3x-1}$ sequence beginning at 84 84, 42, 21, 62, 31, 92, 46, 23, 68, 34, 17, 50, 25, 74, ... A008898 ${\displaystyle 5x-1}$ 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.

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

Logarithms and 21st 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.

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

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

 ${\displaystyle \log _{21}2}$ 0.22767 A152825 ${\displaystyle \log _{2}21}$ 4.39232 A155536 2 21 2097152 ${\displaystyle \log _{21}e}$ 0.328459 ${\displaystyle \log 21}$ 3.04452 A016644 ${\displaystyle e^{21}}$ ${\displaystyle \log _{21}3}$ 0.360849 A153097 ${\displaystyle \log _{3}21}$ 2.77124 A155541 3 21 10460353203 ${\displaystyle \log _{21}\pi }$ 0.375997 ${\displaystyle \log _{\pi }21}$ 2.6596 ${\displaystyle \pi ^{21}}$ ${\displaystyle \log _{21}4}$ 0.45534 A153131 ${\displaystyle \log _{4}21}$ 2.19616 A155545 4 21 4398046511104 ${\displaystyle \log _{21}5}$ 0.528634 A153455 ${\displaystyle \log _{5}21}$ 1.89167 A155553 5 21 476837158203125 ${\displaystyle \log _{21}6}$ 0.588519 A153611 ${\displaystyle \log _{6}21}$ 1.69918 A155554 6 21 21936950640377856 ${\displaystyle \log _{21}7}$ 0.639151 A153632 ${\displaystyle \log _{7}21}$ 1.56458 A155591 7 21 558545864083284007 ${\displaystyle \log _{21}8}$ 0.683011 A153895 ${\displaystyle \log _{8}21}$ 1.46411 A155675 8 21 9223372036854775808 ${\displaystyle \log _{21}9}$ 0.721698 A154020 ${\displaystyle \log _{9}21}$ 1.38562 A155676 9 21 109418989131512359209 ${\displaystyle \log _{21}10}$ 0.756304 A154171 ${\displaystyle \log _{10}21}$ 1.32222 A155677 10 21 1000000000000000000000 ${\displaystyle \log _{21}11}$ 0.78761 A154192 ${\displaystyle \log _{11}21}$ 1.26966 A155678 11 21 7400249944258160101211 ${\displaystyle \log _{21}12}$ 0.816189 A154213 ${\displaystyle \log _{12}21}$ 1.22521 A155679 12 21 46005119909369701466112

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

Values for number theoretic functions with 21 as an argument

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

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

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

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

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

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

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

Factorization of 21 in some quadratic integer rings

Since 21 is composite in ${\displaystyle \mathbb {Z} }$, 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.

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

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

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.

 ${\displaystyle -1}$ 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