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9
9 is the square of 3.
Contents
- 1 Membership in core sequences
- 2 Sequences pertaining to 9
- 3 Partitions of 9
- 4 Roots and powers of 9
- 5 Logarithms and ninth powers
- 6 Values for number theoretic functions with 9 as an argument
- 7 Factorization of some small integers in a quadratic integer ring adjoining ,
- 8 Factorization of 9 in some quadratic integer rings
- 9 Representation of 9 in various bases
- 10 See also
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 | 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 |
sequence beginning at 36 | 36, 18, 9, 26, 13, 38, 19, 56, 28, 14, 7, 20, 10, 5, ... | A008894 |
sequence beginning at 9 | 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, ... | A033479 |
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.
3.00000000 | 9 2 | 81 | ||
2.08008382 | A010581 | 9 3 | 729 | |
1.73205080 | A002194 | 9 4 | 6561 | |
1.55184557 | A011094 | 9 5 | 59049 | |
1.44224957 | A002581 | 9 6 | 531441 | |
1.36873810 | A011261 | 9 7 | 4782969 | |
1.31607401 | A011002 | 9 8 | 43046721 | |
1.27651800 | A011263 | 9 9 | 387420489 | |
1.24573093 | A005532 | 9 10 | 3486784401 | |
1.22109462 | 9 11 | 31381059609 | ||
1.09587269 | 9 12 | 282429536481 | ||
A001019 |
Logarithms and ninth powers
In the OEIS specifically and mathematics in general, refers to the natural logarithm of , whereas all other bases are specified with a subscript.
If is not a multiple of 19, then either or is. Hence the formula for the Legendre symbol .
As above, irrational numbers in the following table are truncated to eight decimal places.
0.31546487 | A152747 | 3.16992500 | A020861 | 2 9 | 512 | |||
0.45511961 | 2.19722457 | A016632 | 8103.08392757 | A091933 | ||||
0.50000000 | A020761 | 2.00000000 | A000038 | 3 9 | 19683 | |||
0.52098902 | 1.91942623 | 29809.09933344 | ||||||
0.63092975 | A102525 | 1.58496250 | A020857 | 4 9 | 262144 | |||
0.73248676 | A153205 | 1.36521238 | A154008 | 5 9 | 1953125 | |||
0.81546487 | A153495 | 1.22629438 | A154009 | 6 9 | 10077696 | |||
0.88562187 | A153619 | 1.12915006 | A113211 | 7 9 | 40353607 | |||
0.94639463 | A153756 | 1.05664166 | A154010 | 8 9 | 134217728 | |||
1.00000000 | 9 9 | 387420489 | ||||||
1.04795163 | A154160 | 0.95424250 | A104139 | 10 9 | 1000000000 |
(See A001017 for the ninth powers of integers).
Values for number theoretic functions with 9 as an argument
0 | ||
–2 | ||
4 | ||
13 | ||
3 | ||
6 | ||
2 | ||
1 | ||
6 | This is the Carmichael lambda function. | |
1 | This is the Liouville lambda function. | |
1.00200839282608221441785... (see A013667) | ||
9! | 362880 | |
40320 |
Factorization of some small integers in a quadratic integer ring adjoining ,
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 . But it has more prime factors in some quadratic integer rings.
3 2 | |||
3 2 | |||
3 2 OR | 3 2 | ||
3 2 | |||
3 2 OR | |||
3 2 | |||
3 2 | |||
3 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
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 |