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# 27

Please do not rely on any information it contains.

27 is a composite number, the cube of 3. Its own cube is 19683. Notice that 1 + 9 + 6 + 8 + 3 = 27; no larger number has this property in base 10.

## Membership in core sequences

 Odd numbers ..., 21, 23, 25, 27, 29, 31, 33, ... A005408 Composite numbers ..., 24, 25, 26, 27, 28, 30, 32, ... A002808 Perfect cubes 1, 8, 27, 64, 125, 216, 343, 512, ... A000578 Powers of 3 1, 3, 9, 27, 81, 243, 729, 2187, ... A000244 Deficient numbers ..., 23, 25, 26, 27, 29, 31, 32, ... A005100

In Pascal's triangle, 27 occurs twice.

## Sequences pertaining to 27

 Multiples of 27 0, 27, 54, 81, 108, 135, 162, 189, 216, 243, 270, 297, 324, ... 27-gonal numbers 1, 27, 78, 154, 255, 381, 532, 708, 909, 1135, 1386, 1662, ... A255186 Decimal expansion of reciprocal of 27 0.037037037037037037037037037037037037037037037... A021031 $3x+1$ sequence beginning at 27 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, ...(takes more than a hundred iterations to get to a power of 2) A008884 $3x-1$ sequence beginning at 27 27, 80, 40, 20, 10, 5, 14, 7, 20, 10, 5, 14, 7, ...

## Partitions of 27

There are 3010 partitions of 27.

There are only two partitions of 27 into squares: 2(1 2) + 5 2 and 3(3 2).

The Goldbach representations of 27 using distinct primes are: 3 + 5 + 19 = 3 + 7 + 17 = 3 + 11 + 13 = 27.

## Roots and powers of 27

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

 ${\sqrt {27}}$ 5.19615242 A010482 27 2 729 ${\sqrt[{3}]{27}}$ 3.00000000 27 3 19683 ${\sqrt[{4}]{27}}$ 2.27950705 A011022 27 4 531441 ${\sqrt[{5}]{27}}$ 1.93318204 A011112 27 5 14348907 ${\sqrt[{6}]{27}}$ 1.73205080 A002194 27 6 387420489 ${\sqrt[{7}]{27}}$ 1.60132888 27 7 10460353203 ${\sqrt[{8}]{27}}$ 1.50980364 27 8 282429536481 ${\sqrt[{9}]{27}}$ 1.44224957 A002581 27 9 7625597484987 ${\sqrt[{10}]{27}}$ 1.39038917 27 10 205891132094649 A009971

## Logarithms and twenty-seventh powers

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

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

TABLE GOES HERE

## Values for number theoretic functions with 27 as an argument

 $\mu (27)$ 0 $M(27)$ –1 $\pi (27)$ 9 $\sigma _{1}(27)$ 40 $\sigma _{0}(27)$ 4 $\phi (27)$ 18 $\Omega (27)$ 3 $\omega (27)$ 1 $\lambda (27)$ 18 This is the Carmichael lambda function. $\lambda (27)$ –1 This is the Liouville lambda function. $\zeta (27)$ 1.0000000074507117898... 27! 10888869450418352160768000000 $\Gamma (27)$ 403291461126605635584000000

## Factorization of some small integers in a quadratic integer ring adjoining the square roots of –27, 27

Since ${\sqrt {27}}=3{\sqrt {3}}$ , $\mathbb {Z} [{\sqrt {27}}]$ is a subdomain of $\mathbb {Z} [{\sqrt {3}}]$ . A similar situation holds for $\mathbb {Z} [{\sqrt {-27}}]$ , but $\mathbb {Z} [{\sqrt {-3}}]$ is not integrally closed either, and it is in fact a subdomain of $\mathbb {Z} [\omega ]$ , where $\omega$ is a complex cubic root of 1.

## Factorization of 27 in some quadratic integer rings

As was mentioned above, 27 is the cube of 3. But to obtain its factorization in quadratic integer rings, it's not always a matter of taking the factorization of 3 and adding exponent 3s as needed.

 $\mathbb {Z} [i]$ 3 3 $\mathbb {Z} [{\sqrt {-2}}]$ $(1-{\sqrt {-2}})^{3}(1-{\sqrt {-2}})^{3}$ $\mathbb {Z} [{\sqrt {2}}]$ 3 3 $\mathbb {Z} [\omega ]$ $(-1)(1+2\omega )^{6}$ $\mathbb {Z} [{\sqrt {3}}]$ $({\sqrt {3}})^{6}$ $\mathbb {Z} [{\sqrt {-5}}]$ 3 3 $\mathbb {Z} [\phi ]$ 3 3 $\mathbb {Z} [{\sqrt {-6}}]$ $\mathbb {Z} [{\sqrt {6}}]$ $(3-{\sqrt {6}})^{3}(3+{\sqrt {6}})^{3}$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}$ $\mathbb {Z} [{\sqrt {7}}]$ $(-1)(2-{\sqrt {7}})^{3}(2+{\sqrt {7}})^{3}$ $\mathbb {Z} [{\sqrt {-10}}]$ $\mathbb {Z} [{\sqrt {10}}]$ 3 3 ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}$ $\left({\frac {1}{2}}-{\frac {\sqrt {-11}}{2}}\right)^{3}\left({\frac {1}{2}}+{\frac {\sqrt {-11}}{2}}\right)^{3}$ $\mathbb {Z} [{\sqrt {11}}]$ $\mathbb {Z} [{\sqrt {-13}}]$ 3 3 ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}$ $\left({\frac {1}{2}}-{\frac {\sqrt {13}}{2}}\right)^{3}\left({\frac {1}{2}}+{\frac {\sqrt {13}}{2}}\right)^{3}$ $\mathbb {Z} [{\sqrt {-14}}]$ $\mathbb {Z} [{\sqrt {14}}]$ 3 3 ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}$ $\mathbb {Z} [{\sqrt {15}}]$ $\mathbb {Z} [{\sqrt {-17}}]$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}$ $\mathbb {Z} [{\sqrt {19}}]$ $(-1)(4-{\sqrt {19}})^{3}(4+{\sqrt {19}})^{3}$ ## Representation of 27 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 11011 1000 123 102 43 36 33 30 27 25 23 21 1D 1C 1B 1A 19 18 17

Note that 27 is a Harshad number in bases 3, 5, 7, 9, and, breaking the pattern of odd bases, 10. Then bases 13 and 19.

 $-1$ 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