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# 20

Please do not rely on any information it contains.

20 is an integer, the number of rooted trees with six vertices (see A000081).

## Membership in core sequences

 Even numbers ..., 14, 16, 18, 20, 22, 24, 26, ... A005843(10) Composite numbers ..., 15, 16, 18, 20, 21, 22, 24, ... A002808 Oblong numbers 2, 6, 12, 20, 30, 42, 56, 72, ... A002378 Tetrahedral numbers 1, 4, 10, 20, 35, 56, 84, 120, ... A000292 Quarter-squares ..., 9, 12, 16, 20, 25, 30, 36, ... A002620 Central binomial coefficients 1, 2, 6, 20, 70, 252, 924, 3432, ... A000984

In Pascal's triangle, 20 occurs thrice, the first time in the sixth row as the sum of 10 and 10 in the row above.

## Core sequences modulo 20

 Integers modulo 20 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, ... Prime numbers modulo 20 2, 3, 5, 7, 11, 13, 17, 19, 3, 9, 11, 17, 1, 3, 7, 13, 19, 1, 7, 11, 13, 19, 3, 9, ... A242120 Fibonacci numbers modulo 20 1, 1, 2, 3, 5, 8, 13, 1, 14, 15, 9, 4, 13, 17, 10, 7, 17, 4, 1, 5, 6, 11, 17, 8, 5, ... A287533 Squares modulo 20 1, 4, 9, 16, 5, 16, 9, 4, 1, 0, 1, 4, 9, 16, 5, 16, 9, 4, 1, 0, 1, 4, 9, 16, 5, 16, ... A070442 Powers of 2 modulo 20 1, 2, 4, 8, 16, 12, 4, 8, 16, 12, 4, 8, 16, 12, 4, 8, 16, 12, 4, 8, 16, 12, 4, 8, ...

## Sequences pertaining to 20

 Multiples of 20 0, 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, ... A008602 20-gonal numbers 1, 20, 57, 112, 185, 276, 385, 512, 657, 820, 1001, 1200, ... A051872 Centered 20-gonal numbers 1, 21, 61, 121, 201, 301, 421, 561, 721, 901, 1101, 1321, ... A069133 ${\displaystyle 3x+1}$ sequence beginning at 15 ..., 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1, ... A033480 ${\displaystyle 3x-1}$ sequence beginning at 5 5, 14, 7, 20, 10, 5, 14, 7, 20, 10, 5, 14, 7, 20, 10, 5, 14, ... A003079 ${\displaystyle 7x-1}$ sequence beginning at 3 3, 20, 10, 5, 34, 17, 118, 59, 412, 206, 103, 720, 360, ... A063871

## Partitions of 20

There are 627 partitions of 20. Of these, only ten are pairs, and of those ten pairs, only two are valid Goldbach representations: 3 + 17 = 7 + 13 = 20.

## Roots and powers of 20

In the table below, irrational numbers are given truncated to eight decimal places.

 ${\displaystyle {\sqrt {20}}}$ 4.47213595 A010476 20 2 400 ${\displaystyle {\sqrt[{3}]{20}}}$ 2.71441761 A010592 20 3 8000 ${\displaystyle {\sqrt[{4}]{20}}}$ 2.11474252 A011016 20 4 160000 ${\displaystyle {\sqrt[{5}]{20}}}$ 1.82056420 A011105 20 5 3200000 ${\displaystyle {\sqrt[{6}]{20}}}$ 1.64754897 A011425 20 6 64000000 ${\displaystyle {\sqrt[{7}]{20}}}$ 1.53412740 A011426 20 7 1280000000 ${\displaystyle {\sqrt[{8}]{20}}}$ 1.45421543 A011427 20 8 25600000000 ${\displaystyle {\sqrt[{9}]{20}}}$ 1.39495079 A011428 20 9 512000000000 ${\displaystyle {\sqrt[{10}]{20}}}$ 1.34928284 A011429 20 10 10240000000000 ${\displaystyle {\sqrt[{11}]{20}}}$ 1.31303243 A011430 20 11 204800000000000 ${\displaystyle {\sqrt[{12}]{20}}}$ 1.28356884 A011431 20 12 4096000000000000 A009964

## Logarithms and twentieth powers

In the table below, irrational numbers are given truncated to eight decimal places.

 ${\displaystyle \log _{20}2}$ 0.231378 A152821 ${\displaystyle \log _{2}20}$ 4.32193 A155172 2 20 1048576 ${\displaystyle \log _{20}e}$ 0.333808 ${\displaystyle \log 20}$ 2.99573 A016643 ${\displaystyle e^{20}}$ ${\displaystyle \log _{20}3}$ 0.366726 A153035 ${\displaystyle \log _{3}20}$ 2.72683 A102447 3 20 3486784401 ${\displaystyle \log _{20}\pi }$ 0.38212 ${\displaystyle \log _{\pi }20}$ 2.61698 ${\displaystyle \pi ^{20}}$ ${\displaystyle \log _{20}4}$ 0.462756 A153124 ${\displaystyle \log _{4}20}$ 2.16096 A155183 4 20 1099511627776 ${\displaystyle \log _{20}5}$ 0.537244 A153454 ${\displaystyle \log _{5}20}$ 1.86135 A155184 5 20 95367431640625 ${\displaystyle \log _{20}6}$ 0.598104 A153610 ${\displaystyle \log _{6}20}$ 1.67195 A155490 6 20 3656158440062976 ${\displaystyle \log _{20}7}$ 0.649561 A153630 ${\displaystyle \log _{7}20}$ 1.5395 A155496 7 20 79792266297612001 ${\displaystyle \log _{20}8}$ 0.694135 A153872 ${\displaystyle \log _{8}20}$ 1.44064 A155502 8 20 1152921504606846976 ${\displaystyle \log _{20}9}$ 0.733452 A154019 ${\displaystyle \log _{9}20}$ 1.36342 A155503 9 20 12157665459056928801 ${\displaystyle \log _{20}10}$ 0.768622 A154170 ${\displaystyle \log _{10}20}$ 1.30103 A155522 10 20 100000000000000000000

(See A010808 for the twentieth powers of integers).

## Values for number theoretic functions with 20 as an argument

 ${\displaystyle \mu (20)}$ 0 ${\displaystyle M(20)}$ −3 ${\displaystyle \pi (20)}$ 8 ${\displaystyle \sigma _{1}(20)}$ 42 ${\displaystyle \sigma _{0}(20)}$ 6 ${\displaystyle \phi (20)}$ 8 ${\displaystyle \Omega (20)}$ 3 ${\displaystyle \omega (20)}$ 2 ${\displaystyle \lambda (20)}$ 4 This is the Carmichael lambda function. ${\displaystyle \lambda (20)}$ −1 This is the Liouville lambda function. ${\displaystyle \zeta (20)}$ 1.000000953962033872796113152... (see A013678). 20! 2432902008176640000 ${\displaystyle \Gamma (20)}$ 121645100408832000

## Factorization of some small integers in a quadratic integer ring adjoining the square roots of −20, 20

Given that 20 is not squarefree, adjoining neither ${\displaystyle {\sqrt {-20}}}$ nor ${\displaystyle {\sqrt {20}}}$ generates an integrally closed ring.

Since ${\displaystyle {\sqrt {-20}}=2{\sqrt {-5}}}$, the form ${\displaystyle a+b{\sqrt {-20}}}$ (with ${\displaystyle a}$ and ${\displaystyle b}$ both integers) skips over numbers of the form ${\displaystyle a+b{\sqrt {-5}}}$ with ${\displaystyle b}$ odd.

This is even more acute for ${\displaystyle {\sqrt {20}}=2{\sqrt {5}}}$, since ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {5}})}=\mathbb {Z} [\phi ]}$, which also includes numbers of the form ${\displaystyle {\frac {a}{2}}+{\frac {b{\sqrt {5}}}{2}}}$ (with ${\displaystyle a}$ and ${\displaystyle b}$ both odd integers).

## Factorization of 20 in some quadratic integer rings

In ${\displaystyle \mathbb {Z} }$, 20 has the prime factorization of 2 2 × 5. But it has different factorizations in some quadratic integer rings.

 ${\displaystyle \mathbb {Z} [i]}$ ${\displaystyle (-i)(1-i)^{2}(1+i)^{2}(2+i)(1+2i)}$ ${\displaystyle \mathbb {Z} [{\sqrt {-2}}]}$ ${\displaystyle (-1)({\sqrt {-2}})^{4}5}$ ${\displaystyle \mathbb {Z} [{\sqrt {2}}]}$ ${\displaystyle ({\sqrt {2}})^{4}5}$ ${\displaystyle \mathbb {Z} [\omega ]}$ 2 2 × 5 ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$ 2 2 × 5 ${\displaystyle \mathbb {Z} [{\sqrt {-5}}]}$ ${\displaystyle (-1)2^{2}({\sqrt {-5}})^{2}}$ ${\displaystyle \mathbb {Z} [\phi ]}$ ${\displaystyle 2^{2}(-1+2\phi )^{2}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-6}}]}$ 2 2 × 5 ${\displaystyle \mathbb {Z} [{\sqrt {6}}]}$ ${\displaystyle (2\pm {\sqrt {6}})^{2}(1\pm {\sqrt {6}})}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}}$ ${\displaystyle \left({\frac {1}{2}}-{\frac {\sqrt {-7}}{2}}\right)^{2}\left({\frac {1}{2}}+{\frac {\sqrt {-7}}{2}}\right)^{2}5}$ ${\displaystyle \mathbb {Z} [{\sqrt {7}}]}$ ${\displaystyle (3-{\sqrt {7}})^{2}(3+{\sqrt {7}})^{2}5}$ ${\displaystyle \mathbb {Z} [{\sqrt {-10}}]}$ 2 2 × 5 ${\displaystyle \mathbb {Z} [{\sqrt {10}}]}$ 2 2 × 5 ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {11}}]}$ ${\displaystyle (3\pm {\sqrt {11}})^{2}(7\pm 2{\sqrt {11}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {-13}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}}$ 2 2 × 5 ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {14}}]}$ ${\displaystyle (-1)(4\pm {\sqrt {14}})^{2}(3\pm {\sqrt {14}})}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {15}}]}$ 2 2 × 5 OR ${\displaystyle 2(5-{\sqrt {15}})(5+{\sqrt {15}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {-17}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}}$ ${\displaystyle \left({\frac {3}{2}}-{\frac {\sqrt {17}}{2}}\right)^{2}\left({\frac {3}{2}}+{\frac {\sqrt {17}}{2}}\right)^{2}5}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}}$ ${\displaystyle 2^{2}\left({\frac {1}{2}}-{\frac {\sqrt {-19}}{2}}\right)\left({\frac {1}{2}}+{\frac {\sqrt {-19}}{2}}\right)}$ ${\displaystyle \mathbb {Z} [{\sqrt {19}}]}$ ${\displaystyle (13\pm 3{\sqrt {19}})^{2}(48\pm 11{\sqrt {19}})}$

## Representation of 20 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 10100 202 110 40 32 26 24 22 20 19 18 17 16 15 14 13 12 11 10

20 is a Harshad number in every base from binary to decimal except for bases 7 and 8.

 ${\displaystyle -1}$ 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