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# 48

Please do not rely on any information it contains.

48 is a composite number, the smallest to have ten divisors (the nontrivial divisors being 2, 3, 4, 6, 8, 12, 16, 24, see A018261).

## Membership in core sequences

 Even numbers ..., 42, 44, 46, 48, 50, 52, 54, ... A005843 Composite numbers ..., 44, 45, 46, 48, 49, 50, 51, ... A002808 Abundant numbers ..., 36, 40, 42, 48, 54, 56, 60, ... A005101 Number of rooted trees with ${\displaystyle n}$ nodes ..., 4, 9, 20, 48, 115, 286, 719, ... A000081

## Sequences pertaining to 48

 Divisors of 48 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 A018261 Multiples of 48 0, 48, 96, 144, 192, 240, 288, 336, 384, 432, 480, 528, 576, ...

## Partitions of 48

There are 147273 partitions of 48. Of these, the of the [FINISH WRITING]

## Roots and powers of 48

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

TABLE GOES HERE

REMARKS

TABLE

TABLE GOES HERE

## Factorization of 48 in some quadratic integer rings

As was mentioned above, 48 is the product of 2 4 and 3. But it has different factorizations in some quadratic integer rings.

 ${\displaystyle \mathbb {Z} [i]}$ ${\displaystyle (1\pm i)^{4}3}$ ${\displaystyle \mathbb {Z} [{\sqrt {-2}}]}$ ${\displaystyle ({\sqrt {-2}})^{8}(1\pm {\sqrt {-2}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {2}}]}$ ${\displaystyle ({\sqrt {2}})^{8}3}$ ${\displaystyle \mathbb {Z} [\omega ]}$ ${\displaystyle (-1)2^{4}(1+2\omega )^{2}}$ ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$ ${\displaystyle (1\pm {\sqrt {3}})^{4}({\sqrt {3}})^{2}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-5}}]}$ 2 4 × 3 ${\displaystyle \mathbb {Z} [\phi ]}$ 2 4 × 3 ${\displaystyle \mathbb {Z} [{\sqrt {-6}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {6}}]}$ ${\displaystyle (2\pm {\sqrt {6}})^{4}(3\pm {\sqrt {6}})}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}}$ ${\displaystyle \left({\frac {1}{2}}\pm {\frac {\sqrt {-7}}{2}}\right)^{4}3}$ ${\displaystyle \mathbb {Z} [{\sqrt {7}}]}$ ${\displaystyle (-1)(3\pm {\sqrt {7}})^{4}(2\pm {\sqrt {7}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {-10}}]}$ 2 4 × 3 ${\displaystyle \mathbb {Z} [{\sqrt {10}}]}$ 2 4 × 3 ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}}$ ${\displaystyle 2^{4}\left({\frac {1}{2}}\pm {\frac {\sqrt {-11}}{2}}\right)}$ ${\displaystyle \mathbb {Z} [{\sqrt {11}}]}$ ${\displaystyle (3\pm {\sqrt {11}})^{3}3}$ ${\displaystyle \mathbb {Z} [{\sqrt {-13}}]}$ 2 4 × 3 ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}}$ ${\displaystyle (-1)2^{4}\left({\frac {1}{2}}\pm {\frac {\sqrt {13}}{2}}\right)}$ ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {14}}]}$ ${\displaystyle (4\pm {\sqrt {14}})^{4}3}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {15}}]}$ 2 4 × 3 ${\displaystyle \mathbb {Z} [{\sqrt {-17}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}}$ ${\displaystyle \left({\frac {3}{2}}\pm {\frac {\sqrt {17}}{2}}\right)^{4}3}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {19}}]}$ ${\displaystyle (-1)(13\pm 3{\sqrt {19}})^{4}(4\pm {\sqrt {19}})}$

## Representation of 48 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 110000 1210 300 143 120 66 60 53 48 44 40 39 36 33 30 2E 2C 2A 28

Notice that 48 is a Harshad number in every base from binary to base 19 except base 14.

 ${\displaystyle -1}$ 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