# 672

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## Number

$672$ (six hundred and seventy-two) is:

$2^5 \times 3 \times 7$

The $2$nd triperfect number after $120$:
$\map {\sigma_1} {672} = 2016 = 3 \times 672$

The $7$th Ore number after $1$, $6$, $28$, $140$, $270$, $496$:
$\dfrac {672 \times \map {\sigma_0} {672} } {\map {\sigma_1} {672} } = 8$
and the $5$th after $1$, $6$, $14$, $270$ whose divisors also have an arithmetic mean which is an integer:
$\dfrac {\map {\sigma_1} {672} } {\map {\sigma_0} {672} } = 84$

The $14$th positive integer after $128$, $192$, $256$, $288$, $320$, $384$, $432$, $448$, $480$, $512$, $576$, $640$, $648$ with $7$ or more prime factors:
$672 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 7$

The $21$st second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $222$, $260$, $301$, $345$, $392$, $442$, $495$, $551$, $610$:
$672 = \dfrac {21 \paren {3 \times 21 + 1} } 2$

The $32$nd Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $216$, $224$, $312$, $315$, $384$, $432$, $612$, $624$:
$672 = 8 \times 84 = 8 \times \paren {6 \times 7 \times 2}$

The $42$nd generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $392$, $425$, $442$, $477$, $495$, $532$, $551$, $590$, $610$, $651$:
$672 = \dfrac {21 \paren {3 \times 21 + 1} } 2$

### Arithmetic Functions on $672$

 $\ds \map {\sigma_0} { 672 }$ $=$ $\ds 24$ $\sigma_0$ of $672$ $\ds \map {\sigma_1} { 672 }$ $=$ $\ds 2016$ $\sigma_1$ of $672$