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# 40

Please do not rely on any information it contains.

40 is an integer. In English, it is the only integer with its letters in alphabetical order (f, o, r, t, y).

## Membership in core sequences

 Even numbers ..., 34, 36, 38, 40, 42, 44, 46, ... A005843(20) Composite numbers ..., 36, 38, 39, 40, 42, 44, 45, ... A002808 Abundant numbers ..., 24, 30, 36, 40, 42, 48, 54, ... A005101 Numbers that are the sum of two squares ..., 34, 36, 37, 40, 41, 45, 49, ... A001481

## Sequences pertaining to 40

 Divisors of 40 1, 2, 4, 5, 8, 10, 20, 40 A018257 Multiples of 40 0, 40, 80, 120, 160, 200, 240, 280, 320, 360, ... Powers of 40 1, 40, 1600, 64000, 2560000, 102400000, ... A009984 ${\displaystyle 3x+1}$ sequence beginning at 39 ..., 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, ... A008878

## Partitions of 40

There are 37338 partitions of 40.

The Goldbach representations of 40 are: 3 + 37 = 11 + 29 = 17 + 23.

## Roots and powers of 40

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

 ${\displaystyle {\sqrt {40}}}$ 6.32455532 A010494 40 2 1600 ${\displaystyle {\sqrt[{3}]{40}}}$ 3.41995189 A010611 40 3 64000 ${\displaystyle {\sqrt[{4}]{40}}}$ 2.51486685 A011034 40 4 2560000 ${\displaystyle {\sqrt[{5}]{40}}}$ 2.09127910 A011125 40 5 102400000 ${\displaystyle {\sqrt[{6}]{40}}}$ 1.84931119 40 6 4096000000 ${\displaystyle {\sqrt[{7}]{40}}}$ 1.69381398 40 7 163840000000 ${\displaystyle {\sqrt[{8}]{40}}}$ 1.58583317 40 8 6553600000000 ${\displaystyle {\sqrt[{9}]{40}}}$ 1.50663019 40 9 262144000000000 ${\displaystyle {\sqrt[{10}]{40}}}$ 1.44612554 40 10 10485760000000000 ${\displaystyle {\sqrt[{11}]{40}}}$ 1.39843349 40 11 419430400000000000 ${\displaystyle {\sqrt[{12}]{40}}}$ 1.35989381 40 12 16777216000000000000 A009984

PLACEHOLDER

PLACEHOLDER

## Factorization of some small integers in a quadratic integer ring adjoining the square roots of −40, 40

Given that 40 is not squarefree, [FINISH WRITING]

## Factorization of 40 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 OR ${\displaystyle (-1)2({\sqrt {-10}})^{2}}$ ${\displaystyle \mathbb {Z} [{\sqrt {10}}]}$ 2 2 × 5 OR ${\displaystyle 2({\sqrt {10}})^{2}}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}}$ 2 2 × 5 ${\displaystyle \mathbb {Z} [{\sqrt {11}}]}$ ${\displaystyle (3\pm {\sqrt {11}})^{2}(7\pm 2{\sqrt {11}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {-13}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}}$ 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 40 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 101000 1111 220 130 104 55 50 44 40 37 34 31 2C 2A 28 26 24 22 20

40 is a Harshad number in every base from binary to vigesimal except for duodecimal, bases 14, 15 and 18.

 ${\displaystyle -1}$ 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