9

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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 $\left(\frac{a}{19}\right)=a^{9}mod19$.

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	$e^9$	8103.08392757	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		${\pi }^9$	29809.09933344
${\log }_{9}4$	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	1953125
${\log }_{9}6$	0.81546487	A153495	${\log }_{6}9$	1.22629438	A154009	6 9	10077696
${\log }_{9}7$	0.88562187	A153619	${\log }_{7}9$	1.12915006	A113211	7 9	40353607
${\log }_{9}8$	0.94639463	A153756	${\log }_{8}9$	1.05664166	A154010	8 9	134217728
${\log }_{9}9$			1.00000000			9 9	387420489
${\log }_{9}10$	1.04795163	A154160	${\log }_{10}9$	0.95424250	A104139	10 9	1000000000

(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
$\varphi (9)$	6
$\mathrm{\Omega }(9)$	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
$\mathrm{\Gamma }(9)$	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 $\mathbb{\mathbb{Z}}$. But it has more prime factors in some quadratic integer rings.

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

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

See also

Some integers

							$-1$
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