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23

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

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

Logarithms and 23rd powers

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

If is not a multiple of 47, 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.

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

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

Values for number theoretic functions with 23 as an argument

–1
–4
9
24
1
22
1
1
22 This is the Carmichael lambda function.
1 This is the Liouville lambda function.
1.0000001192199...
22! 25852016738884976640000
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 , with units of the form (), is a unique factorization domain. , 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.

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

Perhaps it does not need to be said that is not a distinct factorization of 22, since this is a UFD and we readily see that .

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

Factorization of
In In
2
3 Prime
5 Prime
7 Prime
11 Prime
13
17 Prime
19 Prime
23
29
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
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