# 4005

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## Number

$4005$ (four thousand and five) is:

$3^2 \times 5 \times 89$

The $45$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $2145$, $2278$, $2415$, $2556$, $2701$, $2850$, $3003$, $3160$, $3321$, $3486$, $3655$, $3828$:
$4005 = \ds \sum_{k \mathop = 1}^{45} \paren {4 k - 3} = 45 \paren {2 \times 45 - 1}$

The $89$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $3240$, $3321$, $3403$, $3486$, $3570$, $3655$, $3741$, $3828$, $3916$:
$4005 = \ds \sum_{k \mathop = 1}^{89} k = \dfrac {89 \times \paren {89 + 1} } 2$