# 840

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## Number

$840$ (eight hundred and forty) is:

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

The $15$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$:
$\map \tau {840} = 32$

The $15$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$:
$\dfrac {\map \sigma {840} } {840} = \dfrac {2880} {840} \approx 3 \cdotp 429$

The $42$nd highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $\ldots$, $210$, $216$, $240$, $288$, $300$, $336$, $360$, $420$, $480$, $504$, $600$, $630$, $660$, $720$:
$\map \sigma {840} = 2880$

Has the largest $\tau$ value of all the positive integers up to $1000$:
$\map \tau {840} = 32$

### Arithmetic Functions on $840$

 $\displaystyle \map \tau { 840 }$ $=$ $\displaystyle 32$ $\tau$ of $840$ $\displaystyle \map \sigma { 840 }$ $=$ $\displaystyle 2880$ $\sigma$ of $840$