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13
13 is a prime number, considered unlucky by some for reasons that are outside the scope of this reference. However, in a sense, it could be considered doubly lucky, since it survives two different sieving processes: that for prime numbers and that for lucky numbers (see A031157).
Contents
- 1 Membership in core sequences
- 2 Sequences pertaining to 13
- 3 Partitions of 13
- 4 Roots and powers of 13
- 5 Logarithms and thirteenth powers
- 6 Values for number theoretic functions with 13 as an argument
- 7 Factorization of some small integers in a quadratic integer ring adjoining the square roots of −13, 13
- 8 Factorization of 13 in some quadratic integer rings
- 9 Representation of 13 in various bases
- 10 See also
Membership in core sequences
Odd numbers | 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, ... | A005408 |
Prime numbers | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ... | A000040 |
Squarefree numbers | 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, ... | A005117 |
Fibonacci numbers | 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... | A000045 |
Mersenne exponents | 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, ... | A000043 |
In Pascal's triangle, 13 occurs twice. (In Lozanić's triangle, 13 occurs four times).
Sequences pertaining to 13
Multiples of 13 | 0, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, ... | A008595 |
13-gonal numbers | 1, 13, 36, 70, 115, 171, 238, 316, 405, 505, 616, 738, 871, ... | A051865 |
Centered 13-gonal numbers | 1, 14, 40, 79, 131, 196, 274, 365, 469, 586, 716, 859, 1015, ... | A069126 |
Concentric 13-gonal numbers | 1, 13, 27, 52, 79, 117, 157, 208, 261, 325, 391, 468, 547, ... | A195045 |
sequence beginning at 9 | ..., 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 4, 1, 2, ... | A033479 |
sequence beginning at 5 | 5, 26, 13, 66, 33, 166, 83, 416, 208, 104, 52, 26, 13, 66, 33, ... | A259207 |
Number of "Friday the 13ths" in year (starting at 1901) |
2, 1, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 3, 1, 1, 2, 2, 1, 2, ... | A101312 |
13-rough numbers | 1, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, ... | A008365 |
Partitions of 13
There are 101 partitions of 13. Of these, only 18 consist of distinct parts: {13}, {1, 12}, {2, 11}, {3, 10}, {4, 9}, {1, 2, 10}, {1, 3, 9}, {1, 4, 8}, {2, 3, 8}, {1, 5, 7}, {2, 4, 7}, {2, 5, 6}, {3, 4, 6}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 3, 4, 5}. Aside from the trivial partition and {2, 11}, all prime partitions of 13 have repeated parts.
Roots and powers of 13
In the table below, irrational numbers are given truncated to eight decimal places.
3.60555127 | A010470 | 13 2 | 169 | |
2.35133468 | A010585 | 13 3 | 2197 | |
1.89882892 | A011010 | 13 4 | 28561 | |
1.67027765 | A011098 | 13 5 | 371293 | |
1.53340623 | A011320 | 13 6 | 4826809 | |
1.44256291 | A011321 | 13 7 | 62748517 | |
1.37798001 | A011322 | 13 8 | 815730721 | |
1.32975454 | A011323 | 13 9 | 10604499373 | |
1.29239222 | A011324 | 13 10 | 137858491849 | |
1.26260521 | A011325 | 13 11 | 1792160394037 | |
1.23830781 | A011326 | 13 12 | 23298085122481 | |
A001022 |
Logarithms and thirteenth 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.
0.27023815 | A152779 | 3.70043971 | A152590 | 2 13 | 8192 | |||
2.56494935 | A016636 | |||||||
0.42831734 | A153016 | 2.33471751 | A154217 | 3 13 | 1594323 | |||
2903677.27061328 | ||||||||
0.54047630 | A153106 | 1.85021985 | A154224 | 4 13 | 67108864 | |||
0.62747356 | A153313 | 1.59369264 | A154265 | 5 13 | 1220703125 | |||
0.69855549 | A153603 | 1.43152549 | A154278 | 6 13 | 13060694016 | |||
0.75865441 | A153623 | 1.31812322 | A154294 | 7 13 | 96889010407 | |||
0.81071446 | A153855 | 1.23347990 | A154309 | 8 13 | 549755813888 | |||
0.85663468 | A154013 | 1.16735875 | A154339 | 9 13 | 2541865828329 | |||
0.89771171 | A154163 | 1.11394335 | A153496 | 10 13 | 10000000000000 |
See A010801 for the thirteenth powers of integers. In 2004, Reinhard Zumkeller noticed that in base 10, and have the same least significant digit.
Values for number theoretic functions with 13 as an argument
–1 | ||
–3 | ||
6 | ||
14 | ||
2 | ||
12 | ||
1 | ||
1 | ||
12 | This is the Carmichael lambda function. | |
–1 | This is the Liouville lambda function. | |
1.00012271334757848914675... (see A013671). | ||
13! | 6227020800 | |
479001600 |
Factorization of some small integers in a quadratic integer ring adjoining the square roots of −13, 13
is a unique factorization domain, but is not.
2 | Irreducible | Prime |
3 | Prime | |
4 | 2 2 | |
5 | Irreducible | Prime |
6 | 2 × 3 | |
7 | Irreducible | Prime |
8 | 2 3 | |
9 | 3 2 | |
10 | 2 × 5 | |
11 | Irreducible | Prime |
12 | 2 2 × 3 | |
13 | ||
14 | 2 × 7 OR | 2 × 7 |
15 | 3 × 5 | |
16 | 2 4 | |
17 | ||
18 | 2 × 3 2 | |
19 | Irreducible | Prime |
20 | 2 2 × 5 |
Factorization of 13 in some quadratic integer rings
As was mentioned above, 13 is a prime number in . But it is composite in some quadratic integer rings.
Prime | Prime | ||
Prime | |||
Irreducible | Irreducible | ||
Prime | Prime | ||
Irreducible | |||
Irreducible | |||
Prime | Prime |
For through , 13 is prime or irreducible. And beyond , it is always certainly irreducible.
Representation of 13 in various bases
Base | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 through 36 |
Representation | 1101 | 111 | 31 | 23 | 22 | 16 | 15 | 14 | 13 | 12 | 11 | 10 | D |
As the table shows, 13 is a palindromic number in bases 3, 6 and 12.
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 |