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# 26

Please do not rely on any information it contains.

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 ${\displaystyle 3x+1}$ sequence beginning at 9 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, ... A033479 ${\displaystyle 5x+1}$ 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.

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

## Logarithms and 26th 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.

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

 ${\displaystyle \log _{26}2}$ 0.212746 ${\displaystyle \log _{2}26}$ 4.70044 2 26 67108864 ${\displaystyle \log _{26}e}$ 0.306928 ${\displaystyle \log 26}$ 3.2581 A016649 ${\displaystyle e^{26}}$ ${\displaystyle \log _{26}3}$ 0.337195 ${\displaystyle \log _{3}26}$ 2.96565 A152564 3 26 2541865828329 ${\displaystyle \log _{26}\pi }$ 0.351349 ${\displaystyle \log _{\pi }26}$ 2.84617 ${\displaystyle \pi ^{26}}$ ${\displaystyle \log _{26}4}$ 0.425492 ${\displaystyle \log _{4}26}$ 2.35022 4 26 4503599627370496 ${\displaystyle \log _{26}5}$ 0.493981 ${\displaystyle \log _{5}26}$ 2.02437 5 26 1490116119384765625 ${\displaystyle \log _{26}6}$ 0.549941 ${\displaystyle \log _{6}26}$ 1.81838 6 26 170581728179578208256 ${\displaystyle \log _{26}7}$ 0.597254 ${\displaystyle \log _{7}26}$ 1.67433 7 26 9387480337647754305649 ${\displaystyle \log _{26}8}$ 0.638238 ${\displaystyle \log _{8}26}$ 1.56681 8 26 302231454903657293676544 ${\displaystyle \log _{26}9}$ 0.674389 ${\displaystyle \log _{9}26}$ 1.48282 9 26 6461081889226673298932241 ${\displaystyle \log _{26}10}$ 0.706727 ${\displaystyle \log _{10}26}$ 1.41497 10 26 100000000000000000000000000

See A089081 for the 26th powers of integers.

## Values for number theoretic functions with 26 as an argument

 ${\displaystyle \mu (26)}$ 1 ${\displaystyle M(26)}$ −3 ${\displaystyle \pi (26)}$ 9 ${\displaystyle \sigma _{1}(26)}$ 42 ${\displaystyle \sigma _{0}(26)}$ 4 ${\displaystyle \phi (26)}$ 12 ${\displaystyle \Omega (26)}$ 2 ${\displaystyle \omega (26)}$ 2 ${\displaystyle \lambda (26)}$ 12 This is the Carmichael lambda function. ${\displaystyle \lambda (26)}$ 1 This is the Liouville lambda function. ${\displaystyle \zeta (26)}$ 1.0000000149015548... 26! 403291461126605635584000000 ${\displaystyle \Gamma (26)}$ 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 ${\displaystyle \scriptstyle \mathbb {Z} [{\sqrt {26}}]\,}$, with units of the form ${\displaystyle \scriptstyle \pm (5+{\sqrt {26}})^{n}\,}$ (${\displaystyle \scriptstyle n\,\in \,\mathbb {Z} \,}$), is not a unique factorization domain, having class number 2. ${\displaystyle \mathbb {Z} [{\sqrt {-26}}]}$ 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.

 ${\displaystyle n}$ ${\displaystyle \mathbb {Z} [{\sqrt {-26}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {26}}]}$ 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 ${\displaystyle (-1)(4-{\sqrt {26}})(4+{\sqrt {26}})}$ 11 Prime Irreducible 12 2 2 × 3 13 Irreducible 14 2 × 7 15 3 × 5 16 2 4 17 Irreducible ${\displaystyle (11-2{\sqrt {26}})(11+2{\sqrt {26}})}$ 18 2 × 3 2 19 Prime Irreducible 20 2 2 × 5 21 3 × 7 22 2 × 11 2 × 11 OR ${\displaystyle (-1)(2-{\sqrt {26}})(2+{\sqrt {26}})}$ 23 Prime ${\displaystyle (-1)(9-2{\sqrt {26}})(9+2{\sqrt {26}})}$ 24 2 3 × 3 25 5 2 5 2 OR ${\displaystyle (-1)(1-{\sqrt {26}})(1+{\sqrt {26}})}$ 26 2 × 13 OR ${\displaystyle (-1)({\sqrt {-26}})^{2}}$ 2 × 13 OR ${\displaystyle ({\sqrt {26}})^{2}}$ 27 3 3 OR ${\displaystyle (1-{\sqrt {-26}})(1+{\sqrt {-26}})}$ 3 3 28 2 2 × 7 29 Prime 30 2 × 3 × 5 OR ${\displaystyle (2-{\sqrt {-26}})(2+{\sqrt {-26}})}$ 2 × 3 × 5

To drive home the point that ${\displaystyle \mathbb {Z} [{\sqrt {-26}}]}$ 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.

 ${\displaystyle n}$ ${\displaystyle \mathbb {Z} [{\sqrt {-26}}]}$ 42 2 × 3 × 7 OR ${\displaystyle (4-{\sqrt {-26}})(4+{\sqrt {-26}})}$ 75 3 × 5 2 OR ${\displaystyle (7-{\sqrt {-26}})(7+{\sqrt {-26}})}$ 90 2 × 3 2 × 5 OR ${\displaystyle (8-{\sqrt {-26}})(8+{\sqrt {-26}})}$ 105 3 × 5 × 7 OR ${\displaystyle (2-{\sqrt {-26}})(2+{\sqrt {-26}})}$ 108 2 2 × 3 3 OR ${\displaystyle (2-2{\sqrt {-26}})(2+2{\sqrt {-26}})}$ 120 2 3 × 3 × 5 OR ${\displaystyle (4-2{\sqrt {-26}})(4+2{\sqrt {-26}})}$ 126 2 2 × 5 2 OR ${\displaystyle (10-{\sqrt {-26}})(10+{\sqrt {-26}})}$

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

 ${\displaystyle p}$ Factorization of ${\displaystyle \langle p\rangle }$ In ${\displaystyle \mathbb {Z} [{\sqrt {-26}}]}$ In ${\displaystyle \mathbb {Z} [{\sqrt {26}}]}$ 2 ${\displaystyle \langle 2,{\sqrt {-26}}\rangle ^{2}}$ ${\displaystyle \langle 2,{\sqrt {26}}\rangle ^{2}}$ 3 ${\displaystyle \langle 3,1-{\sqrt {-26}}\rangle \langle 3,1+{\sqrt {-26}}\rangle }$ Prime 5 ${\displaystyle \langle 5,2-{\sqrt {-26}}\rangle \langle 5,2+{\sqrt {-26}}\rangle }$ ${\displaystyle \langle 5,1-{\sqrt {26}}\rangle \langle 5,1+{\sqrt {26}}\rangle }$ 7 ${\displaystyle \langle 7,3-{\sqrt {-26}}\rangle \langle 7,3+{\sqrt {-26}}\rangle }$ Prime 11 Prime ${\displaystyle \langle 11,2-{\sqrt {26}}\rangle \langle 11,2+{\sqrt {26}}\rangle }$ 13 ${\displaystyle \langle 13,{\sqrt {-26}}\rangle ^{2}}$ ${\displaystyle \langle 13,{\sqrt {26}}\rangle ^{2}}$ 17 ${\displaystyle \langle 17,5-{\sqrt {-26}}\rangle \langle 17,5+{\sqrt {-26}}\rangle }$ ${\displaystyle \langle 3-{\sqrt {26}}\rangle \langle 3+{\sqrt {26}}\rangle }$ 19 Prime ${\displaystyle \langle 19,8-{\sqrt {26}}\rangle \langle 19,8+{\sqrt {26}}\rangle }$ 23 ${\displaystyle \langle 9-2{\sqrt {26}}\rangle \langle 9+2{\sqrt {26}}\rangle }$ 29 Prime 31 ${\displaystyle \langle 31,6-{\sqrt {-26}}\rangle \langle 31,6+{\sqrt {-26}}\rangle }$ Prime 37 ${\displaystyle \langle 37,14-{\sqrt {-26}}\rangle \langle 37,14+{\sqrt {-26}}\rangle }$ ${\displaystyle \langle 37,10-{\sqrt {26}}\rangle \langle 37,10+{\sqrt {26}}\rangle }$ 41 Prime Prime 43 ${\displaystyle \langle 43,19-{\sqrt {-26}}\rangle \langle 43,19+{\sqrt {-26}}\rangle }$ 47 ${\displaystyle \langle 47,16-{\sqrt {-26}}\rangle \langle 47,16+{\sqrt {-26}}\rangle }$

## Factorization of 26 in some quadratic integer rings

As was mentioned above, 26 is the product of two primes in ${\displaystyle \mathbb {Z} }$. But it has different factorizations in some quadratic integer rings.

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

Surprisingly enough, ${\displaystyle ({\sqrt {26}})^{2}}$ is a distinct factorization of 26 in ${\displaystyle \mathbb {Z} [{\sqrt {26}}]}$, since this is not a UFD and we readily see that ${\displaystyle {\sqrt {26}}}$ 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.

 ${\displaystyle -1}$ 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