54

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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

Values for number theoretic functions with 54 as an argument

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.

See also

Some integers

							$-1$
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