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# 18

Please do not rely on any information it contains.

18 is an integer, the only positive integer that is twice the sum of its base 10 digits (see A169805).

## Membership in core sequences

 Even numbers ..., 12, 14, 16, 18, 20, 22, 24, ... A005843 Composite numbers ..., 14, 15, 16, 18, 20, 21, 22, ... A002808 Lucas numbers 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ... A000032 Abundant numbers 12, 18, 20, 24, 30, 36, 40, 42, ... A005101 Numbers that are the sum of two squares ..., 13, 16, 17, 18, 20, 25, 26, ... A001481

## Sequences pertaining to 18

 Multiples of 18 0, 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216, ... A008600 Octadecagonal numbers 1, 18, 51, 100, 165, 246, 343, 456, 585, 730, 891, 1068, ... A051870 $3x-1$ sequence starting at 36 36, 18, 9, 26, 13, 38, 19, 56, 28, 14, 7, 20, 10, 5, 14, 7, ... A008894 $5x+1$ sequence starting at 11 11, 56, 28, 14, 7, 36, 18, 9, 46, 23, 116, 58, 29, 146, 73, ... A259193

## Partitions of 18

There are 385 partitions of 18.

The Goldbach representations of 18 are: 13 + 5 = 11 + 7 = 18.

## Roots and powers of 18

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

 ${\sqrt {18}}$ 4.24264068 A010474 18 2 324 ${\sqrt[{3}]{18}}$ 2.62074139 A010590 18 3 5832 ${\sqrt[{4}]{18}}$ 2.05976714 A011014 18 4 104976 ${\sqrt[{5}]{18}}$ 1.78260245 A011103 18 5 1889568 ${\sqrt[{6}]{18}}$ 1.61887040 A011395 18 6 34012224 ${\sqrt[{7}]{18}}$ 1.51120939 A011396 18 7 612220032 ${\sqrt[{8}]{18}}$ 1.43518888 A011397 18 8 11019960576 ${\sqrt[{9}]{18}}$ 1.37871570 A011398 18 9 198359290368 ${\sqrt[{10}]{18}}$ 1.33514136 A011399 18 10 3570467226624 ${\sqrt[{11}]{18}}$ 1.30051594 A011400 18 11 64268410079232 ${\sqrt[{12}]{18}}$ 1.27234838 A011401 18 12 1156831381426176 A001027

## Logarithms and eighteenth 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.

If $n$ is not a multiple of 37, then either $n^{18}-1$ or $n^{18}+1$ is. Hence the formula for the Legendre symbol $\left({\frac {a}{37}}\right)=a^{18}\mod 37$ .

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

TABLE GOES HERE

(See A010806 for the eighteenth powers of integers).

## Values for number theoretic functions with 18 as an argument

 $\mu (18)$ 0 $M(18)$ −2 $\pi (18)$ 7 $\sigma _{1}(18)$ 39 $\sigma _{0}(18)$ 18 $\phi (18)$ 6 $\Omega (18)$ 3 $\omega (18)$ 2 $\lambda (18)$ 6 This is the Carmichael lambda function. $\lambda (18)$ −1 This is the Liouville lambda function. $\zeta (18)={\frac {43867\pi ^{18}}{38979295480125}}$ = 1.000003817293... See A013676. 18! 6402373705728000 $\Gamma (18)$ 355687428096000

## Factorization of 18 in some quadratic integer rings

As was mentioned above, 18 is composite in $\mathbb {Z}$ . But it has different factorizations in some quadratic integer rings.

 $\mathbb {Z} [i]$ $(1\pm i)3^{2}$ $\mathbb {Z} [{\sqrt {-2}}]$ $(-1)({\sqrt {-2}})^{2}(1\pm {\sqrt {-2}})^{2}$ $\mathbb {Z} [{\sqrt {2}}]$ $({\sqrt {2}})^{2}3^{2}$ $\mathbb {Z} [\omega ]$ $(-1)2(1+2\omega )^{4}$ $\mathbb {Z} [{\sqrt {3}}]$ $(-1)(1\pm {\sqrt {3}})({\sqrt {3}})^{4}$ $\mathbb {Z} [{\sqrt {-5}}]$ 2 × 3 2 $\mathbb {Z} [\phi ]$ 2 × 3 2 $\mathbb {Z} [{\sqrt {-6}}]$ $\mathbb {Z} [{\sqrt {6}}]$ $(-1)(2\pm {\sqrt {6}})(3\pm {\sqrt {6}})^{2}$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}$ $\left({\frac {1}{2}}\pm {\frac {\sqrt {-7}}{2}}\right)3^{2}$ $\mathbb {Z} [{\sqrt {7}}]$ $(3\pm {\sqrt {7}})(2\pm {\sqrt {7}})^{2}$ $\mathbb {Z} [{\sqrt {-10}}]$ 2 × 3 2 $\mathbb {Z} [{\sqrt {10}}]$ 2 × 3 2 ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}$ $2\left({\frac {1}{2}}\pm {\frac {\sqrt {-11}}{2}}\right)^{2}$ $\mathbb {Z} [{\sqrt {11}}]$ $(-1)(3\pm {\sqrt {11}})3^{2}$ $\mathbb {Z} [{\sqrt {-13}}]$ 2 × 3 2 ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}$ $2\left({\frac {1}{2}}\pm {\frac {\sqrt {13}}{2}}\right)^{2}$ $\mathbb {Z} [{\sqrt {-14}}]$ $\mathbb {Z} [{\sqrt {14}}]$ $(4\pm {\sqrt {14}})3^{2}$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}$ $\mathbb {Z} [{\sqrt {15}}]$ 2 × 3 2 $\mathbb {Z} [{\sqrt {-17}}]$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}$ $(-1)\left({\frac {3}{2}}\pm {\frac {\sqrt {17}}{2}}\right)3^{2}$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}$ $\mathbb {Z} [{\sqrt {19}}]$ $(-1)(13\pm 3{\sqrt {19}})(4\pm {\sqrt {19}})^{2}$ ## Representation of 18 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 through 36 Representation 10010 200 102 33 30 24 22 20 18 17 16 15 14 13 12 11 10 I

18 is a Harshad number in every base from binary to decimal except for octal. It is also a Harshad number in base 13, hexadecimal, base 17 and base 18.

 $-1$ 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