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# 17

Please do not rely on any information it contains.

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 ${\displaystyle 3x+1}$ sequence starting at 9 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, ... A033479 ${\displaystyle 3x-1}$ 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.

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

## Logarithms and seventeenth powers

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

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

 ${\displaystyle \log _{17}2}$ 0.244651 A152782 ${\displaystyle \log _{2}17}$ 4.08746 A154847 2 17 131072 ${\displaystyle \log _{17}e}$ 0.352956 ${\displaystyle \log 17}$ 2.83321 A016640 ${\displaystyle e^{17}}$ 24154952.75357529 ${\displaystyle \log _{17}3}$ 0.387762 A153020 ${\displaystyle \log _{3}17}$ 2.33472 A154217 3 17 129140163 ${\displaystyle \log _{17}\pi }$ 0.404039 ${\displaystyle \log _{\pi }17}$ 2.47501 ${\displaystyle \pi ^{17}}$ ${\displaystyle \log _{17}4}$ 0.489301 A153109 ${\displaystyle \log _{4}17}$ 2.04373 A154849 4 17 17179869184 ${\displaystyle \log _{17}5}$ 0.568061 A153430 ${\displaystyle \log _{5}17}$ 1.76037 A154850 5 17 762939453125 ${\displaystyle \log _{17}6}$ 0.632412 A153607 ${\displaystyle \log _{6}17}$ 1.58125 A154856 6 17 16926659444736 ${\displaystyle \log _{17}7}$ 0.686821 A153627 ${\displaystyle \log _{7}17}$ 1.45598 A154857 7 17 232630513987207 ${\displaystyle \log _{17}8}$ 0.733952 A153858 ${\displaystyle \log _{8}17}$ 1.36249 A154858 8 17 2251799813685248 ${\displaystyle \log _{17}9}$ 0.775524 A154016 ${\displaystyle \log _{9}17}$ 1.28945 A154859 9 17 16677181699666569 ${\displaystyle \log _{17}10}$ 0.812712 A154167 ${\displaystyle \log _{10}17}$ 1.23045 A154860 10 17 100000000000000000

See A010805 for the seventeenth powers of integers.

## Values for number theoretic functions with 17 as an argument

 ${\displaystyle \mu (17)}$ −1 ${\displaystyle M(17)}$ −2 ${\displaystyle \pi (17)}$ 7 ${\displaystyle \sigma _{1}(17)}$ 18 ${\displaystyle \sigma _{0}(17)}$ 2 ${\displaystyle \phi (17)}$ 16 ${\displaystyle \Omega (17)}$ 1 ${\displaystyle \omega (17)}$ 1 ${\displaystyle \lambda (17)}$ 16 This is the Carmichael lambda function. ${\displaystyle \lambda (17)}$ −1 This is the Liouville lambda function. ${\displaystyle \zeta (17)}$ 1.0000076371976... (see A013675). 17! 355687428096000 ${\displaystyle \Gamma (17)}$ 20922789888000

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

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

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

## Factorization of 17 in some quadratic integer rings

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

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

## 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

 ${\displaystyle -1}$ 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