# 720

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## Number

$720$ (seven hundred and twenty) is:

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

The $6$th factorial after $1$, $2$, $6$, $24$, $120$:
$720 = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$

The $14$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$:
$\map \tau {720} = 24$

The $14$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$:
$\dfrac {\map \sigma {720} } {720} = \dfrac {2418} {720} = 3 \cdotp 358 \dot 3$

The $16$th positive integer after $128$, $192$, $256$, $288$, $320$, $384$, $432$, $448$, $480$, $512$, $576$, $640$, $648$, $672$, $704$ with $7$ or more prime factors:
$720 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5$

The $41$st highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $\ldots$, $210$, $216$, $240$, $288$, $300$, $336$, $360$, $420$, $480$, $504$, $600$, $630$, $660$:
$\map \sigma {720} = 2418$

The product of consecutive integers in $2$ different ways:
$720 = 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8$

The only factorial which can be expressed as the sum of two squares:
$6! = 12^2 + 24^2$

### Arithmetic Functions on $720$

 $\ds \map \tau { 720 }$ $=$ $\ds 24$ $\tau$ of $720$ $\ds \map \sigma { 720 }$ $=$ $\ds 2418$ $\sigma$ of $720$