# 1680

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## Number

$1680$ (one thousand, six hundred and eighty) is:

$2^4 \times 3 \times 5 \times 7$

The $17$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$, $840$, $1260$:
$\map {\sigma_0} {1680} = 40$

The $17$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$, $840$, $1260$:
$\dfrac {\map {\sigma_1} {1680} } {1680} = \dfrac {5952} {1680} \approx 3 \cdotp 543$

The $24$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $645$, $736$, $833$, $936$, $1045$, $1160$, $1281$, $1408$, $1541$:
$1680 = \ds \sum_{k \mathop = 1}^{24} \paren {6 k - 5} = 24 \paren {3 \times 24 - 2}$

### Arithmetic Functions on $1680$

 $\ds \map {\sigma_0} { 1680 }$ $=$ $\ds 40$ $\sigma_0$ of $1680$ $\ds \map {\sigma_1} { 1680 }$ $=$ $\ds 5952$ $\sigma_1$ of $1680$