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# 54

Please do not rely on any information it contains.

54 is an integer, the smallest that can be written as a sum of three squares in three different ways: 7 2 + 2 2 + 1 2 = 6 2 + 2(3 2) = 2(5 2) + 2 2 = 54.

## Membership in core sequences

 Even numbers ..., 48, 50, 52, 54, 56, 58, 60, ... A005843 Composite numbers ..., 50, 51, 52, 54, 55, 56, 57, ... A002808 Abundant numbers ..., 40, 42, 48, 54, 56, 60, 66, ... A005101

## Sequences pertaining to 54

 Multiples of 54 0, 54, 108, 162, 216, 270, 324, 378, 432, 486, 540, 594, 648, ...

## Partitions of 54

There are 386155 partitions of 54.

The Goldbach representations of 54 are: 7 + 47 = 11 + 43 = 13 + 41 = 17 + 37 = 23 + 31.

## Roots and powers of 54

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

PLACEHOLDER

PLACEHOLDER

## Factorization of 54 in some quadratic integer rings

As was mentioned above, 54 is composite in $\mathbb {Z}$ . But it has different factorizations in some quadratic integer rings.

 $\mathbb {Z} [i]$ $(1\pm i)3^{3}$ $\mathbb {Z} [{\sqrt {-2}}]$ $(-1)({\sqrt {-2}})^{2}(1\pm {\sqrt {-2}})^{3}$ $\mathbb {Z} [{\sqrt {2}}]$ $({\sqrt {2}})^{2}3^{3}$ $\mathbb {Z} [\omega ]$ $(-1)2(1+2\omega )^{6}$ $\mathbb {Z} [{\sqrt {3}}]$ $(-1)(1\pm {\sqrt {3}})({\sqrt {3}})^{6}$ $\mathbb {Z} [{\sqrt {-5}}]$ 2 × 3 3 $\mathbb {Z} [\phi ]$ 2 × 3 3 $\mathbb {Z} [{\sqrt {-6}}]$ $\mathbb {Z} [{\sqrt {6}}]$ $(-1)(2\pm {\sqrt {6}})(3\pm {\sqrt {6}})^{2}$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}$ $\left({\frac {1}{2}}\pm {\frac {\sqrt {-7}}{2}}\right)3^{3}$ $\mathbb {Z} [{\sqrt {7}}]$ $(-1)(3\pm {\sqrt {7}})(2\pm {\sqrt {7}})^{3}$ $\mathbb {Z} [{\sqrt {-10}}]$ 2 × 3 3 $\mathbb {Z} [{\sqrt {10}}]$ 2 × 3 3 ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}$ $2\left({\frac {1}{2}}\pm {\frac {\sqrt {-11}}{2}}\right)^{3}$ $\mathbb {Z} [{\sqrt {11}}]$ $(-1)(3\pm {\sqrt {11}})3^{3}$ $\mathbb {Z} [{\sqrt {-13}}]$ 2 × 3 3 ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}$ $(-1)2\left({\frac {1}{2}}\pm {\frac {\sqrt {13}}{2}}\right)^{3}$ $\mathbb {Z} [{\sqrt {-14}}]$ $\mathbb {Z} [{\sqrt {14}}]$ $(4\pm {\sqrt {14}})3^{3}$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}$ $\mathbb {Z} [{\sqrt {15}}]$ 2 × 3 3 $\mathbb {Z} [{\sqrt {-17}}]$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}$ $(-1)\left({\frac {3}{2}}\pm {\frac {\sqrt {17}}{2}}\right)3^{3}$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}$ $\mathbb {Z} [{\sqrt {19}}]$ $(13\pm 3{\sqrt {19}})(4\pm {\sqrt {19}})^{3}$ ## Representation of 54 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 110110 2000 312 204 130 105 66 60 54 4A 46 42 3C 39 36 33 30 2G 2E

54 is a Harshad number in bases 3, 4, 5, 7, 9, 10, 13, 16, 17, 18, 19.

 $-1$ 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