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# 9

Please do not rely on any information it contains.

9 is the square of 3.

## Membership in core sequences

 Odd numbers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, ... A005408 Composite numbers 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, ... A002808 Perfect squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ... A000290 Powers of 3 1, 3, 9, 27, 81, 243, 729, 2187, 6561, ... A000244

In Pascal's triangle, 9 occurs twice. (In Lozanić's triangle, 9 occurs six times, the first two instances surrounding 10 in the central column).

## Sequences pertaining to 9

 Multiples of 9 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, ... A008591 9's complement of $n$ 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 89, 88, 87, 86, 85, 84, ... A061601 Nonagonal numbers 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, ... A001106 $3x-1$ sequence beginning at 36 36, 18, 9, 26, 13, 38, 19, 56, 28, 14, 7, 20, 10, 5, ... A008894 $3x+1$ sequence beginning at 9 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, ... A033479 $5x+1$ sequence beginning at 11 11, 56, 28, 14, 7, 36, 18, 9, 46, 23, 116, 58, 29, ... A259193

## Partitions of 9

There are thirty partitions of 9.

There is only one Goldbach representation of 9 using distinct primes: 2 + 7 = 9.

## Roots and powers of 9

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

 ${\sqrt {9}}$ 3.00000000 9 2 81 ${\sqrt[{3}]{9}}$ 2.08008382 A010581 9 3 729 ${\sqrt[{4}]{9}}$ 1.73205080 A002194 9 4 6561 ${\sqrt[{5}]{9}}$ 1.55184557 A011094 9 5 59049 ${\sqrt[{6}]{9}}$ 1.44224957 A002581 9 6 531441 ${\sqrt[{7}]{9}}$ 1.36873810 A011261 9 7 4782969 ${\sqrt[{8}]{9}}$ 1.31607401 A011002 9 8 43046721 ${\sqrt[{9}]{9}}$ 1.27651800 A011263 9 9 387420489 ${\sqrt[{10}]{9}}$ 1.24573093 A005532 9 10 3486784401 ${\sqrt[{11}]{9}}$ 1.22109462 9 11 31381059609 ${\sqrt[{12}]{9}}$ 1.09587269 9 12 282429536481 A001019

## Logarithms and ninth 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 19, then either $n^{9}-1$ or $n^{9}+1$ is. Hence the formula for the Legendre symbol $\left({\frac {a}{19}}\right)=a^{9}\mod 19$ .

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

 $\log _{9}2$ 0.31546487 A152747 $\log _{2}9$ 3.16992500 A020861 2 9 512 $\log _{9}e$ 0.45511961 $\log 9$ 2.19722457 A016632 $e^{9}$ 8103.08 A091933 $\log _{9}3$ 0.50000000 A020761 $\log _{3}9$ 2.00000000 A000038 3 9 19683 $\log _{9}\pi$ 0.52098902 $\log _{\pi }9$ 1.91942623 $\pi ^{9}$ 29809.1 $\log _{9}4$ 0.63092975 A102525 $\log _{4}9$ 1.58496250 A020857 4 9 262144 $\log _{9}5$ 0.73248676 A153205 $\log _{5}9$ 1.36521238 A154008 5 9 1.95312e+06 $\log _{9}6$ 0.81546487 A153495 $\log _{6}9$ 1.22629438 A154009 6 9 1.00777e+07 $\log _{9}7$ 0.88562187 A153619 $\log _{7}9$ 1.12915006 A113211 7 9 4.03536e+07 $\log _{9}8$ 0.94639463 A153756 $\log _{8}9$ 1.05664166 A154010 8 9 1.34218e+08 $\log _{9}9$ 1.00000000 9 9 3.8742e+08 $\log _{9}10$ 1.04795163 A154160 $\log _{10}9$ 0.95424250 A104139 10 9 1e+09

(See A001017 for the ninth powers of integers).

## Values for number theoretic functions with 9 as an argument

 $\mu (9)$ 0 $M(9)$ –2 $\pi (9)$ 4 $\sigma _{1}(9)$ 13 $\sigma _{0}(9)$ 3 $\phi (9)$ 6 $\Omega (9)$ 2 $\omega (9)$ 1 $\lambda (9)$ 6 This is the Carmichael lambda function. $\lambda (9)$ 1 This is the Liouville lambda function. $\zeta (9)$ 1.00200839282608221441785... (see A013667) 9! 362880 $\Gamma (9)$ 40320

## Factorization of some small integers in a quadratic integer ring adjoining ${\sqrt {-9}}$ , ${\sqrt {9}}$ Since 9 is the square of 3, [FINISH WRITING]

## Factorization of 9 in some quadratic integer rings

As was mentioned above, 9 is the square of 3 and has only one distinct prime factor in $\mathbb {Z}$ . But it has more prime factors in some quadratic integer rings.

 $\mathbb {Z} [i]$ 3 2 $\mathbb {Z} [{\sqrt {-2}}]$ $(1-{\sqrt {-2}})^{2}(1-{\sqrt {-2}})^{2}$ $\mathbb {Z} [{\sqrt {2}}]$ 3 2 $\mathbb {Z} [\omega ]$ $(1+2\omega )^{4}$ $\mathbb {Z} [{\sqrt {3}}]$ $({\sqrt {3}})^{4}$ $\mathbb {Z} [{\sqrt {-5}}]$ 3 2 OR $(2-{\sqrt {-5}})(2+{\sqrt {-5}})$ $\mathbb {Z} [\phi ]$ 3 2 $\mathbb {Z} [{\sqrt {-6}}]$ 3 2 $\mathbb {Z} [{\sqrt {6}}]$ $(3-{\sqrt {6}})^{2}(3+{\sqrt {6}})^{2}$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}$ $\mathbb {Z} [{\sqrt {7}}]$ $(2-{\sqrt {7}})^{2}(2+{\sqrt {7}})^{2}$ $\mathbb {Z} [{\sqrt {-10}}]$ $\mathbb {Z} [{\sqrt {10}}]$ 3 2 OR $(-1)(1-{\sqrt {10}})(1+{\sqrt {10}})$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}$ $\left({\frac {1}{2}}-{\frac {\sqrt {-11}}{2}}\right)^{2}\left({\frac {1}{2}}+{\frac {\sqrt {-11}}{2}}\right)^{2}$ $\mathbb {Z} [{\sqrt {11}}]$ 3 2 $\mathbb {Z} [{\sqrt {-13}}]$ 3 2 ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}$ $\left({\frac {1}{2}}-{\frac {\sqrt {13}}{2}}\right)^{2}\left({\frac {1}{2}}+{\frac {\sqrt {13}}{2}}\right)^{2}$ $\mathbb {Z} [{\sqrt {-14}}]$ $\mathbb {Z} [{\sqrt {14}}]$ 3 2 ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}$ $\mathbb {Z} [{\sqrt {15}}]$ $\mathbb {Z} [{\sqrt {-17}}]$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}$ $\mathbb {Z} [{\sqrt {19}}]$ $(4-{\sqrt {19}})^{2}(4+{\sqrt {19}})^{2}$ ## Representation of 9 in various bases

 Base 2 3 4 5 6 7 8 9 10 through 36 Representation 1001 100 21 14 13 12 11 10 9

 $-1$ 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