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# 13

Please do not rely on any information it contains.

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).

## 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 ${\displaystyle 3x+1}$ sequence beginning at 9 ..., 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 4, 1, 2, ... A033479 ${\displaystyle 5x+1}$ 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 ${\displaystyle n}$(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.

 ${\displaystyle {\sqrt {13}}}$ 3.60555127 A010470 13 2 169 ${\displaystyle {\sqrt[{3}]{13}}}$ 2.35133468 A010585 13 3 2197 ${\displaystyle {\sqrt[{4}]{13}}}$ 1.89882892 A011010 13 4 28561 ${\displaystyle {\sqrt[{5}]{13}}}$ 1.67027765 A011098 13 5 371293 ${\displaystyle {\sqrt[{6}]{13}}}$ 1.53340623 A011320 13 6 4826809 ${\displaystyle {\sqrt[{7}]{13}}}$ 1.44256291 A011321 13 7 62748517 ${\displaystyle {\sqrt[{8}]{13}}}$ 1.37798001 A011322 13 8 815730721 ${\displaystyle {\sqrt[{9}]{13}}}$ 1.32975454 A011323 13 9 10604499373 ${\displaystyle {\sqrt[{10}]{13}}}$ 1.29239222 A011324 13 10 137858491849 ${\displaystyle {\sqrt[{11}]{13}}}$ 1.26260521 A011325 13 11 1792160394037 ${\displaystyle {\sqrt[{12}]{13}}}$ 1.23830781 A011326 13 12 23298085122481 A001022

## Logarithms and thirteenth 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.

 ${\displaystyle \log _{13}2}$ 0.27023815 A152779 ${\displaystyle \log _{2}13}$ 3.70043971 A152590 2 13 8192 ${\displaystyle \log _{13}e}$ ${\displaystyle \log 13}$ 2.56494935 A016636 ${\displaystyle e^{13}}$ ${\displaystyle \log _{13}3}$ 0.42831734 A153016 ${\displaystyle \log _{3}13}$ 2.33471751 A154217 3 13 1594323 ${\displaystyle \log _{13}\pi }$ ${\displaystyle \log _{\pi }13}$ ${\displaystyle \pi ^{13}}$ 2903677.27061328 ${\displaystyle \log _{13}4}$ 0.54047630 A153106 ${\displaystyle \log _{4}13}$ 1.85021985 A154224 4 13 67108864 ${\displaystyle \log _{13}5}$ 0.62747356 A153313 ${\displaystyle \log _{5}13}$ 1.59369264 A154265 5 13 1220703125 ${\displaystyle \log _{13}6}$ 0.69855549 A153603 ${\displaystyle \log _{6}13}$ 1.43152549 A154278 6 13 13060694016 ${\displaystyle \log _{13}7}$ 0.75865441 A153623 ${\displaystyle \log _{7}13}$ 1.31812322 A154294 7 13 96889010407 ${\displaystyle \log _{13}8}$ 0.81071446 A153855 ${\displaystyle \log _{8}13}$ 1.23347990 A154309 8 13 549755813888 ${\displaystyle \log _{13}9}$ 0.85663468 A154013 ${\displaystyle \log _{9}13}$ 1.16735875 A154339 9 13 2541865828329 ${\displaystyle \log _{13}10}$ 0.89771171 A154163 ${\displaystyle \log _{10}13}$ 1.11394335 A153496 10 13 10000000000000

See A010801 for the thirteenth powers of integers. In 2004, Reinhard Zumkeller noticed that in base 10, ${\displaystyle n}$ and ${\displaystyle n^{13}}$ have the same least significant digit.

## Values for number theoretic functions with 13 as an argument

 ${\displaystyle \mu (13)}$ –1 ${\displaystyle M(13)}$ –3 ${\displaystyle \pi (13)}$ 6 ${\displaystyle \sigma _{1}(13)}$ 14 ${\displaystyle \sigma _{0}(13)}$ 2 ${\displaystyle \phi (13)}$ 12 ${\displaystyle \Omega (13)}$ 1 ${\displaystyle \omega (13)}$ 1 ${\displaystyle \lambda (13)}$ 12 This is the Carmichael lambda function. ${\displaystyle \lambda (13)}$ –1 This is the Liouville lambda function. ${\displaystyle \zeta (13)}$ 1.00012271334757848914675... (see A013671). 13! 6227020800 ${\displaystyle \Gamma (13)}$ 479001600

## Factorization of some small integers in a quadratic integer ring adjoining the square roots of −13, 13

${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}}$ is a unique factorization domain, but ${\displaystyle \mathbb {Z} [{\sqrt {-13}}]}$ is not.

 ${\displaystyle n}$ ${\displaystyle \mathbb {Z} [{\sqrt {-13}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}}$ 2 Irreducible Prime 3 Prime ${\displaystyle \left({\frac {5}{2}}-{\frac {\sqrt {13}}{2}}\right)\left({\frac {5}{2}}+{\frac {\sqrt {13}}{2}}\right)}$ 4 2 2 5 Irreducible Prime 6 2 × 3 ${\displaystyle 2\left({\frac {5}{2}}-{\frac {\sqrt {13}}{2}}\right)\left({\frac {5}{2}}+{\frac {\sqrt {13}}{2}}\right)}$ 7 Irreducible Prime 8 2 3 9 3 2 ${\displaystyle \left({\frac {5}{2}}-{\frac {\sqrt {13}}{2}}\right)^{2}\left({\frac {5}{2}}+{\frac {\sqrt {13}}{2}}\right)^{2}}$ 10 2 × 5 11 Irreducible Prime 12 2 2 × 3 ${\displaystyle 2^{2}\left({\frac {5}{2}}-{\frac {\sqrt {13}}{2}}\right)\left({\frac {5}{2}}+{\frac {\sqrt {13}}{2}}\right)}$ 13 ${\displaystyle (-1)({\sqrt {-13}})^{2}}$ ${\displaystyle ({\sqrt {13}})^{2}}$ 14 2 × 7 OR ${\displaystyle (1-{\sqrt {-13}})(1+{\sqrt {-13}})}$ 2 × 7 15 3 × 5 ${\displaystyle \left({\frac {5}{2}}-{\frac {\sqrt {13}}{2}}\right)\left({\frac {5}{2}}+{\frac {\sqrt {13}}{2}}\right)5}$ 16 2 4 17 ${\displaystyle (2-{\sqrt {-13}})(2+{\sqrt {-13}})}$ ${\displaystyle \left({\frac {9}{2}}-{\frac {\sqrt {13}}{2}}\right)\left({\frac {9}{2}}+{\frac {\sqrt {13}}{2}}\right)}$ 18 2 × 3 2 ${\displaystyle 2\left({\frac {5}{2}}-{\frac {\sqrt {13}}{2}}\right)^{2}\left({\frac {5}{2}}+{\frac {\sqrt {13}}{2}}\right)^{2}}$ 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 ${\displaystyle \mathbb {Z} }$. But it is composite in some quadratic integer rings.

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

For ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ through ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-163}})}}$, 13 is prime or irreducible. And beyond ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-163}})}}$, 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.

 ${\displaystyle -1}$ 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