# 216

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## Number

$216$ (two hundred and sixteen) is:

$2^3 \times 3^3$

The $1$st cube which can be expressed as the sum of $3$ positive cubes:
$216 = 3^3 + 4^3 + 5^3$

The $1$st element of the $2$nd set of $4$ positive integers which form an arithmetic sequence which all have the same Euler $\phi$ value:
$\map \phi {216} = \map \phi {222} = \map \phi {228} = \map \phi {234} = 72$

The $3$rd power of $6$ after $(1)$, $6$, $36$:
$216 = 6^3$

The $6$th cube number after $1$, $8$, $27$, $64$, $125$:
$216 = 6 \times 6 \times 6$

The $7$th positive integer after $64$, $96$, $128$, $144$, $160$, $192$ with $6$ or more prime factors:
$216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3$

The $10$th inconsummate number after $62$, $63$, $65$, $75$, $84$, $95$, $161$, $173$, $195$:
$\nexists n \in \Z_{>0}: n = 216 \times \map {s_{10} } n$

The $12$th of the $17$ positive integers for which the value of the Euler $\phi$ function is $72$:
$73$, $91$, $95$, $111$, $117$, $135$, $146$, $148$, $152$, $182$, $190$, $216$, $222$, $228$, $234$, $252$, $270$

The $13$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $146$, $162$, $188$, $206$, $210$

The $24$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$, $24$, $36$, $111$, $112$, $115$, $128$, $132$, $135$, $144$, $175$, $212$:
$216 = 18 \times 12 = 18 \times \paren {2 \times 1 \times 6}$

The $24$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $108$, $121$, $125$, $128$, $144$, $169$, $196$, $200$

The $28$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$, $72$, $84$, $90$, $96$, $108$, $120$, $144$, $168$, $180$, $210$:
$\map {\sigma_1} {216} = 600$

The $42$nd positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.

## Historical Note

$216$ is one of the two numbers generally believed to be Plato's geometrical number:

... which has control over the good and evil of births.
-- Plato's Republic: $546$