17

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17 is a prime number. It is the number of wallpaper groups.

Membership in core sequences

Odd numbers	..., 11, 13, 15, 17, 19, 21, 23, ...	A005408
Prime numbers	..., 7, 11, 13, 17, 19, 23, 29, ...	A000040
Squarefree numbers	..., 13, 14, 15, 17, 19, 21, 22, ...	A005117
Mersenne exponents	2, 3, 5, 7, 13, 17, 19, 31, 61, ...	A000043

In Pascal's triangle, 17 occurs twice. (In Lozanić's triangle, 17 occurs four times).

Sequences pertaining to 17

Multiples of 17	0, 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, ...	A008599
17-gonal numbers	1, 17, 48, 94, 155, 231, 322, 428, 549, 685, 836, 1002, ...	A051869
$3x+1$ sequence starting at 9	9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, ...	A033479
$3x-1$ sequence starting at 17	17, 50, 25, 74, 37, 110, 55, 164, 82, 41, 122, 61, 182, ...	A003124
17-rough numbers	1, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...	A008366

Partitions of 17

There are 297 partitions of 17.

The Goldbach representations of 17 are: 2 + 2 + 13 = 3 + 3 + 11 = 3 + 7 + 7 = 5 + 5 + 7 = 17.

Roots and powers of 17

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

$\sqrt{17}$	4.12310562	A010473	17 2	289
$\sqrt[3]{17}$	2.57128159	A010589	17 3	4913
$\sqrt[4]{17}$	2.03054318	A011013	17 4	83521
$\sqrt[5]{17}$	1.76234034	A011102	17 5	1419857
$\sqrt[6]{17}$	1.60352162	A011380	17 6	24137569
$\sqrt[7]{17}$	1.49891987	A011381	17 7	410338673
$\sqrt[8]{17}$	1.42497129	A011382	17 8	6975757441
$\sqrt[9]{17}$	1.36998731	A011383	17 9	118587876497
$\sqrt[10]{17}$	1.32753167	A011384	17 10	2015993900449
				A001026

Logarithms and seventeenth 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.

${\log }_{17}2$	0.24465054	A152782	${\log }_{2}17$	4.08746284	A154847	2 17	131072
${\log }_{17}e$	0.35295612		$\log 17$	2.83321334	A016640	$e^{17}$	24154952.75357529
${\log }_{17}3$	0.38776193	A153020	${\log }_{3}17$	2.33471751	A154217	3 17	129140163
${\log }_{17}\pi$	0.40403942		${\log }_{\pi }17$	2.47500600		${\pi }^{17}$
${\log }_{17}4$	0.48930108	A153109	${\log }_{4}17$	2.04373142	A154849	4 17	17179869184
${\log }_{17}5$	0.56806096	A153430	${\log }_{5}17$	1.76037442	A154850	5 17	762939453125
${\log }_{17}6$	0.63241247	A153607	${\log }_{6}17$	1.58124647	A154856	6 17	16926659444736
${\log }_{17}7$	0.68682090	A153627	${\log }_{7}17$	1.45598364	A154857	7 17	232630513987207
${\log }_{17}8$	0.73395162	A153858	${\log }_{8}17$	1.36248761	A154858	8 17	2251799813685248
${\log }_{17}9$	0.77552387	A154016	${\log }_{9}17$	1.28945096	A154859	9 17	16677181699666569
${\log }_{17}10$	0.81271150	A154167	${\log }_{10}17$	1.23044892	A154860	10 17	100000000000000000

See A010805 for the seventeenth powers of integers.

Values for number theoretic functions with 17 as an argument

$\mu (17)$	−1
$M(17)$	−2
$\pi (17)$	7
${\sigma }_1(17)$	18
${\sigma }_0(17)$	2
$\varphi (17)$	16
$\mathrm{\Omega }(17)$	1
$\omega (17)$	1
$\lambda (17)$	16	This is the Carmichael lambda function.
$\lambda (17)$	−1	This is the Liouville lambda function.
$\zeta (17)$	1.0000076371976... (see A013675).
17!	355687428096000
$\mathrm{\Gamma }(17)$	20922789888000

Factorization of some small integers in quadratic integer rings adjoining the square roots of −17, 17

${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{17})}$ is a unique factorization domain, but $\mathbb{\mathbb{Z}}[\sqrt{-17}]$ is not. Units in ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{17})}$ are of the form $(4+\sqrt{17}{)}^n$.

$n$	$\mathbb{\mathbb{Z}}[\sqrt{-17}]$	${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{17})}$
2	Irreducible	$(\frac{5}{2}-\frac{\sqrt{17}}{2})(\frac{5}{2}+\frac{\sqrt{17}}{2})$
3		Prime
4	2 2	${(\frac{5}{2}-\frac{\sqrt{17}}{2})}^2{(\frac{5}{2}+\frac{\sqrt{17}}{2})}^2$
5	Prime
6	2 × 3	$(\frac{5}{2}-\frac{\sqrt{17}}{2})(\frac{5}{2}+\frac{\sqrt{17}}{2})3$
7	Irreducible	Prime
8	2 3	${(\frac{5}{2}-\frac{\sqrt{17}}{2})}^3{(\frac{5}{2}+\frac{\sqrt{17}}{2})}^3$
9	3 2
10	2 × 5	$(\frac{5}{2}-\frac{\sqrt{17}}{2})(\frac{5}{2}+\frac{\sqrt{17}}{2})5$
11	Irreducible	Prime
12	2 2 × 3	${(\frac{5}{2}-\frac{\sqrt{17}}{2})}^2{(\frac{5}{2}+\frac{\sqrt{17}}{2})}^{2}3$
13	Irreducible	$(-1)(2-\sqrt{17})(2+\sqrt{17})$
14	2 × 7	$(\frac{5}{2}-\frac{\sqrt{17}}{2})(\frac{5}{2}+\frac{\sqrt{17}}{2})7$
15	3 × 5	$(\frac{5}{2}-\frac{\sqrt{17}}{2})(\frac{5}{2}+\frac{\sqrt{17}}{2})5$
16	2 4	${(\frac{5}{2}-\frac{\sqrt{17}}{2})}^4{(\frac{5}{2}+\frac{\sqrt{17}}{2})}^4$
17	$(-1)(\sqrt{-17}{)}^2$	$(\sqrt{17}{)}^2$
18	2 × 3 2 OR $(1-\sqrt{-17})(1+\sqrt{-17})$	$(\frac{5}{2}-\frac{\sqrt{17}}{2})(\frac{5}{2}+\frac{\sqrt{17}}{2})3$
19	Irreducible	Prime
20	2 2 × 5	${(\frac{5}{2}-\frac{\sqrt{17}}{2})}^2{(\frac{5}{2}+\frac{\sqrt{17}}{2})}^{2}5$
21	3 × 7 OR $(2-\sqrt{-17})(2+\sqrt{-17})$	3 × 7

Factorization of 17 in some quadratic integer rings

As was mentioned above, 17 is a prime number in $\mathbb{\mathbb{Z}}$. But it is composite in some quadratic integer rings.

$\mathbb{\mathbb{Z}}[i]$	$4+i$
$\mathbb{\mathbb{Z}}[\sqrt{-2}]$	$(3-2\sqrt{-2})(3+2\sqrt{-2})$	$\mathbb{\mathbb{Z}}[\sqrt{2}]$	$(-1)(1-3\sqrt{2})(1+3\sqrt{2})$
$\mathbb{\mathbb{Z}}[\omega ]$	Prime	$\mathbb{\mathbb{Z}}[\sqrt{3}]$	Prime
$\mathbb{\mathbb{Z}}[\sqrt{-5}]$	Irreducible	$\mathbb{\mathbb{Z}}[\varphi ]$
$\mathbb{\mathbb{Z}}[\sqrt{-6}]$		$\mathbb{\mathbb{Z}}[\sqrt{6}]$
${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-7})}$	Prime	$\mathbb{\mathbb{Z}}[\sqrt{7}]$
$\mathbb{\mathbb{Z}}[\sqrt{-10}]$	Irreducible	$\mathbb{\mathbb{Z}}[\sqrt{10}]$	Irreducible
${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-11})}$	Prime	$\mathbb{\mathbb{Z}}[\sqrt{11}]$	Prime
$\mathbb{\mathbb{Z}}[\sqrt{-13}]$	$(2-\sqrt{-13})(2+\sqrt{-13})$	${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{13})}$	$(\frac{9}{2}-\frac{\sqrt{13}}{2})(\frac{9}{2}+\frac{\sqrt{13}}{2})$
$\mathbb{\mathbb{Z}}[\sqrt{-14}]$	Irreducible	$\mathbb{\mathbb{Z}}[\sqrt{14}]$	Prime
${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-15})}$		$\mathbb{\mathbb{Z}}[\sqrt{15}]$	Irreducible despite positive Legendre symbol
$\mathbb{\mathbb{Z}}[\sqrt{-17}]$	$(-1)(\sqrt{-17}{)}^2$	${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{17})}$	$(\sqrt{17}{)}^2$
${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-19})}$	Prime	$\mathbb{\mathbb{Z}}[\sqrt{19}]$	$(6-\sqrt{19})(6+\sqrt{19})$

Representation of 17 in various bases

Base	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18 through 36
Representation	10001	122	101	32	25	23	21	18	17	16	15	14	13	12	11	10	H

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