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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
sequence starting at 9 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, ... A033479
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.

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

Logarithms and seventeenth powers

In the OEIS specifically and mathematics in general, refers to the natural logarithm of , whereas all other bases are specified with a subscript.

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

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

See A010805 for the seventeenth powers of integers.

Values for number theoretic functions with 17 as an argument

−1
−2
7
18
2
16
1
1
16 This is the Carmichael lambda function.
−1 This is the Liouville lambda function.
1.0000076371976... (see A013675).
17! 355687428096000
20922789888000

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

is a unique factorization domain, but is not. Units in are of the form .

2 Irreducible
3 Prime
4 2 2
5 Prime
6 2 × 3
7 Irreducible Prime
8 2 3
9 3 2
10 2 × 5
11 Irreducible Prime
12 2 2 × 3
13 Irreducible
14 2 × 7
15 3 × 5
16 2 4
17
18 2 × 3 2 OR
19 Irreducible Prime
20 2 2 × 5
21 3 × 7 OR 3 × 7

Factorization of 17 in some quadratic integer rings

As was mentioned above, 17 is a prime number in . But it is composite in some quadratic integer rings.

Prime Prime
Irreducible
Prime
Irreducible Irreducible
Prime Prime
Irreducible Prime
Irreducible despite positive Legendre symbol
Prime

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
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