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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
sequence beginning at 21 21, 64, 32, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, ... A033481
sequence beginning at 84 84, 42, 21, 62, 31, 92, 46, 23, 68, 34, 17, 50, 25, 74, ... A008898
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.

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

Logarithms and 21st powers

In the OEIS specifically and mathematics in general, refers to the natural logarithm of , 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 is not a multiple of 43, then either or is. Hence the formula for the Legendre symbol .

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

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

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

Values for number theoretic functions with 21 as an argument

1
–2
8
32
4
12
2
2
6 This is the Carmichael lambda function.
1 This is the Liouville lambda function.
1.000000476932986787806463116719604373...
21! 51090942171709440000
2432902008176640000

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

is a unique factorization domain, but is not. The fundamental unit in is , with norm 1.

2 Irreducible Prime
3
4 2 2
5 Irreducible
6 2 × 3
7 Irreducible
8 2 3
9 3 2
10 2 × 5
11 Irreducible Prime
12 2 2 × 3
13 Prime
14 2 × 7
15 3 × 5
16 2 4
17 Irreducible
18 2 × 3 2
19 Irreducible Prime
20 2 2 × 5
21 3 × 7 OR

It is worth emphasizing that is not a distinct factorization of 21. Observe that .

Ideals really help us make sense of multiple distinct factorizations in .

Factorization of
In In
2 Prime
3
5
7
11 Prime
13 Prime
17
19 Prime
23
29
31
37
41
43
47

Factorization of 21 in some quadratic integer rings

Since 21 is composite in , 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.

3 × 7
3 × 7 OR OR 3 × 7
3 × 7 3 × 7
3 × 7
3 × 7 OR
3 × 7 OR 3 × 7

Of particular interest, note how 21 has three distinct factorizations in .

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
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