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# 216

Please do not rely on any information it contains.

216 is the cube of 6, and is the smallest cube that can be written as a sum of three positive cubes, specifically: 3 3 + 4 3 + 5 3 = 6 3.

## Membership in core sequences

 Even numbers ..., 210, 212, 214, 216, 218, 220, 222, ... A005843 Composite numbers ..., 213, 214, 215, 216, 217, 218, 219, ... A002808 Abundant numbers ..., 204, 208, 210, 216, 220, 222, 224, ... A005101 Cubes 1, 8, 27, 64, 125, 216, 343, 512, 729, ... A000578

## Sequences pertaining to 216

 Multiples of 216 216, 432, 648, 864, 1080, 1296, 1512, 1728, 1944, 2160, 2376, 2592, ... Divisors of 216 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216 A018338

## Partitions of 216

There are 15285151248481 partitions of 216.

The Goldbach representations of 216 are 211 + 5 = 199 + 17 = 197 + 19 = 193 + 23 = 179 + 37 = 173 + 43 = 163 + 53 = 157 + 59 = 149 + 67 = 137 + 79 = 127 + 89 = 113 + 103 = 109 + 107 = 216. Notice in that last one that 107 and 109 are successive primes (see A001043 for more numbers that share this property with 216).

There are 170 subsets of the distinct divisors of 216 that add up to 216 (see A033630). These range from the trivial single element subset to more elaborate partitions like 1 + 2 + 3 + 4 + 8 + 9 + 12 + 18 + 24 + 27 + 108 = 216.

## Roots and powers of 216

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

TABLE GOES HERE

TABLE GOES HERE

## Factorization of 216 in some quadratic integer rings

As was mentioned above, 216 is the cube of 6, which in ${\displaystyle \mathbb {Z} }$ factorizes as 2 × 3. Therefore the prime factorization of 216 is 2 3 × 3 3. But in some quadratic integer rings, either 2 or 3, or both, can be further factorized.

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

## Representation of 216 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 11011000 22000 3120 1331 1000 426 330 260 216 187 160 138 116 E6 D8 CC C0 B7 AG

 ${\displaystyle -1}$ 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