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26

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26 is the only integer to be directly between a square (25) and a cube (27).

Membership in core sequences

Even numbers ..., 20, 22, 24, 26, 28, 30, 32, ... A005843(13)
Composite numbers ..., 22, 24, 25, 26, 27, 28, 30, ... A002808
Semiprimes ..., 21, 22, 25, 26, 33, 34, 35, ... A001358
Squarefree numbers ..., 21, 22, 23, 26, 29, 30, 31, ... A005117
Numbers that are the sum of two squares ..., 18, 20, 25, 26, 29, 32, 34, ... A001481
Young tableaux numbers ..., 2, 4, 10, 26, 76, 232, 764, ... A000085

In Pascal's triangle, 26 occurs twice.

Sequences pertaining to 26

Multiples of 26 0, 26, 52, 78, 104, 130, 156, 182, 208, 234, 260, ... A252994
Decimal expansion of reciprocal of 26 0.03846153846153846153846153846153846153846... A021030
26-gonal numbers 1, 26, 75, 148, 245, 366, 511, 680, 873, 1090, 1331, ... A255185
sequence beginning at 9 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, ... A033479
sequence beginning at 5 5, 26, 13, 66, 33, 166, 83, 416, 208, 104, 52, 26, 13, ... A259207

Partitions of 26

There are 2436 partitions of 26.

The Goldbach representations of 26 are 3 + 23 = 7 + 19 = 13 + 13.

Roots and powers of 26

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

5.09901951 A010481 26 2 676
2.96249606 A010598 26 3 17576
2.25810086 A011021 26 4 456976
1.91864519 A011111 26 5 11881376
1.72119030 26 6 308915776
1.59271859 26 7 8031810176
1.50269786 26 8 208827064576
1.43621434 26 9 5429503678976
1.38515168 26 10 141167095653376
A009970

Logarithms and 26th powers

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

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

0.21274605 4.70043971 2 26 67108864
0.30692767 3.25809653 A016649
0.33719451 2.96564727 A152564 3 26 2541865828329
0.35134928 2.84617059
0.42549210 2.35021985 4 26 4503599627370496
0.49398103 2.02436919 5 26 1490116119384765625
0.54994057 1.81837830 6 26 170581728179578208256
0.59725368 1.67433041 7 26 9387480337647754305649
0.63823816 1.56681323 8 26 302231454903657293676544
0.67438903 1.48282363 9 26 6461081889226673298932241
0.70672709 1.41497334 10 26 100000000000000000000000000

See A089081 for the 26th powers of integers.

Values for number theoretic functions with 26 as an argument

1
−3
9
42
4
12
2
2
12 This is the Carmichael lambda function.
1 This is the Liouville lambda function.
1.0000000149015548...
26! 403291461126605635584000000
15511210043330985984000000

Factorization of some small integers in a quadratic integer ring adjoining the square roots of −26, 26

The commutative quadratic integer ring with unity , with units of the form (), is not a unique factorization domain, having class number 2. is not a unique factorization domain either, though the lack of unique factorization could be said to be "much worse" with a class number of 6.

2 Irreducible Irreducible
3 Prime
4 2 2
5 Irreducible
6 2 × 3
7 Irreducible Prime
8 2 3
9 3 2
10 2 × 5 2 × 5 OR
11 Prime Irreducible
12 2 2 × 3
13 Irreducible
14 2 × 7
15 3 × 5
16 2 4
17 Irreducible
18 2 × 3 2
19 Prime Irreducible
20 2 2 × 5
21 3 × 7
22 2 × 11 2 × 11 OR
23 Prime
24 2 3 × 3
25 5 2 5 2 OR
26 2 × 13 OR 2 × 13 OR
27 3 3 OR 3 3
28 2 2 × 7
29 Prime
30 2 × 3 × 5 OR 2 × 3 × 5

To drive home the point that has class number 6, we'll show a few more numbers which not only have more than one distinct factorization, but the distinct factorizations have a different number of irreducible factors.

42 2 × 3 × 7 OR
75 3 × 5 2 OR
90 2 × 3 2 × 5 OR
105 3 × 5 × 7 OR
108 2 2 × 3 3 OR
120 2 3 × 3 × 5 OR
126 2 2 × 5 2 OR

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

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

Factorization of 26 in some quadratic integer rings

As was mentioned above, 26 is the product of two primes in . But it has different factorizations in some quadratic integer rings.

2 × 13
2 × 13
2 × 13 2 × 13
2 × 13
2 × 13

Surprisingly enough, is a distinct factorization of 26 in , since this is not a UFD and we readily see that is not divisible by either 2 or 13.

Representation of 26 in various bases

Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Representation 11010 222 122 101 42 35 32 28 26 24 22 20 1C 1B 1A 19 18 17 16

As you can see from the table, 26 is palindromic in bases 3, 5 and 12, and also base 25, and trivially base 27 and higher. Its square, 676, palindromic in bases 5, 10, 11, 12, 25. See A002778 for more numbers having a square that is palindromic in base 10.

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