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# 64

Please do not rely on any information it contains.

64 is the square of 8.

## Membership in core sequences

 Even numbers ..., 58, 60, 62, 64, 66, 68, 70, ... A005843 Composite numbers ..., 60, 62, 63, 64, 65, 66, 68, ... A002808 Cubes 1, 8, 27, 64, 125, 216, 343, 512, ... A000578 Powers of 2 1, 2, 4, 8, 16, 32, 64, 128, 256, ... A000079

In Pascal's triangle, 64 occurs only twice, namely in row 64, in the second and next to last positions. But it also appears in a subtler way, as the sum of the numbers in row 6: 1 + 6 15 + 20 + 15 + 6 + 1 = 64.

## Sequences pertaining to 64

 Multiples of 64 0, 64, 128, 192, 256, 320, 384, 448, 512, ... A152691 ${\displaystyle 3x-1}$ sequence beginning at 87 ..., 86, 43, 128, 64, 32, 16, 8, 4, 2, 1, 2, ... A008899 ${\displaystyle 3x+1}$ sequence beginning at 21 21, 64, 32, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, ... A033481

## Partitions of 64

There are 1741630 partitions of 64.

The Goldbach representations of 64 are: 3 + 61 = 5 + 59 = 11 + 53 = 17 + 47 = 23 + 41.

## Roots and powers of 64

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

 ${\displaystyle {\sqrt[{2}]{64}}}$ 8.00000000 64 2 4096 ${\displaystyle {\sqrt[{3}]{64}}}$ 4.00000000 64 3 262144 ${\displaystyle {\sqrt[{4}]{64}}}$ 2.82842712 A010466 64 4 16777216 ${\displaystyle {\sqrt[{5}]{64}}}$ 2.29739670 A011149 64 5 1073741824 ${\displaystyle {\sqrt[{6}]{64}}}$ 2.00000000 64 6 68719476736 ${\displaystyle {\sqrt[{7}]{64}}}$ 1.81144732 A011246 64 7 4398046511104 ${\displaystyle {\sqrt[{8}]{64}}}$ 1.68179283 A011006 64 8 281474976710656 ${\displaystyle {\sqrt[{9}]{64}}}$ 1.58740105 A005480 64 9 18014398509481984 ${\displaystyle {\sqrt[{10}]{64}}}$ 1.51571656 A011093 64 10 1152921504606846976 A089357

Of course the roots given above are the principal real roots. There are also negative real roots and complex roots.

• 8, –8 (both real)
• 4, ${\displaystyle -2\pm {\sqrt {-12}}}$ (the two complex roots are the same except for the sign of the imaginary part)
• ${\displaystyle {\sqrt[{4}]{64}}}$, ${\displaystyle -{\sqrt[{4}]{64}}}$, ${\displaystyle i{\sqrt[{4}]{64}}}$, ${\displaystyle -i{\sqrt[{4}]{64}}}$
• ${\displaystyle {\sqrt[{5}]{64}}}$, etc.

## Logarithms and 64th 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.

TABLE GOES HERE

TABLE GOES HERE

## Factorization of 64 in some quadratic integer rings

In ${\displaystyle \mathbb {Z} }$, 64 has the prime factorization of 2 6. But it has different factorizations in some quadratic integer rings.

 ${\displaystyle \mathbb {Z} [i]}$ ${\displaystyle (1-i)^{6}(1+i)^{6}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-2}}]}$ ${\displaystyle ({\sqrt {-2}})^{12}}$ ${\displaystyle \mathbb {Z} [{\sqrt {2}}]}$ ${\displaystyle ({\sqrt {2}})^{12}}$ ${\displaystyle \mathbb {Z} [\omega ]}$ 2 6 ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$ ${\displaystyle (1-{\sqrt {3}})^{6}(1+{\sqrt {3}})^{6}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-5}}]}$ ${\displaystyle \mathbb {Z} [\phi ]}$ 2 6 ${\displaystyle \mathbb {Z} [{\sqrt {-6}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {6}}]}$ ${\displaystyle (2-{\sqrt {6}})^{6}(2+{\sqrt {6}})^{6}}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}}$ ${\displaystyle \left({\frac {1}{2}}-{\frac {\sqrt {-7}}{2}}\right)^{6}\left({\frac {1}{2}}+{\frac {\sqrt {-7}}{2}}\right)^{6}}$ ${\displaystyle \mathbb {Z} [{\sqrt {7}}]}$ ${\displaystyle (3-{\sqrt {7}})^{6}(3+{\sqrt {7}})^{6}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-10}}]}$ 2 6 ${\displaystyle \mathbb {Z} [{\sqrt {10}}]}$ 2 6 ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {11}}]}$ ${\displaystyle (3-{\sqrt {11}})^{6}(3+{\sqrt {11}})^{6}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-13}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}}$ 2 6 ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {14}}]}$ ${\displaystyle (4-{\sqrt {14}})^{6}(4+{\sqrt {14}})^{6}}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}}$ 2 6 OR ${\displaystyle \left({\frac {1}{2}}-{\frac {\sqrt {-15}}{2}}\right)^{3}\left({\frac {1}{2}}+{\frac {\sqrt {-15}}{2}}\right)^{3}}$ ${\displaystyle \mathbb {Z} [{\sqrt {15}}]}$ 2 6 ${\displaystyle \mathbb {Z} [{\sqrt {-17}}]}$ 2 6 ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}}$ ${\displaystyle \left({\frac {3}{2}}-{\frac {\sqrt {17}}{2}}\right)^{6}\left({\frac {3}{2}}+{\frac {\sqrt {17}}{2}}\right)^{6}}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {19}}]}$ ${\displaystyle (13-3{\sqrt {19}})^{6}(13+3{\sqrt {19}})^{6}}$

## Representation of 64 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 1000000 2101 1000 224 144 121 100 71 64 59 54 4C 48 44 40 3D 3A 37 34

## References

 ${\displaystyle -1}$ 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