# 378

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## Number

$378$ (three hundred and seventy-eight) is:

$2 \times 3^3 \times 7$

The first of the $3$rd pair of triangular numbers whose sum and difference are also both triangular:
$378 = T_{27}$, $703 = T_{37}$, $378 + 703 = T_{46}$, $703 - 378 = T_{25}$

The total of all the entries in a magic cube of order $3$, after $1$, $(36)$:
$378 = \ds \sum_{k \mathop = 1}^{3^3} k = \dfrac {3^3 \paren {3^3 + 1} } 2$

The $14$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $120$, $153$, $190$, $231$, $276$, $325$:
$378 = \ds \sum_{k \mathop = 1}^{14} \paren {4 k - 3} = 14 \paren {2 \times 14 - 1}$

The $15$th Smith number after $4$, $22$, $27$, $58$, $85$, $94$, $121$, $166$, $202$, $265$, $274$, $319$, $346$, $355$:
$3 + 7 + 8 = 2 + 3 + 3 + 3 + 7 = 18$

The $27$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $171$, $190$, $210$, $231$, $253$, $276$, $300$, $325$, $351$:
$378 = \ds \sum_{k \mathop = 1}^{27} k = \dfrac {27 \times \paren {27 + 1} } 2$