714

Number
$714$ (seven hundred and fourteen) is:
 * $2 \times 3 \times 7 \times 17$


 * The $1$st of the $5$th (and largest known) pair of consecutive integers whose product is a primorial:
 * $714 \times 715 = 510 \, 510 = 17 \#$


 * The $8$th integer after $1$, $14$, $30$, $105$, $248$, $264$, $418$ whose divisor sum divided by its Euler $\phi$ value is a square:
 * $\dfrac {\map {\sigma_1} {714} } {\map \phi {714} } = \dfrac {1728} {192} = 9 = 3^2$


 * The $10$th positive integer after $1$, $7$, $102$, $110$, $142$, $159$, $187$, $381$, $690$ the sum of whose divisors is a cube:
 * $\map {\sigma_1} {714} = 1728 = 12^3$


 * The $21$st integer $n$ after $1$, $3$, $15$, $30$, $35$, $56$, $\ldots$, $210$, $248$, $264$, $357$, $420$, $570$, $616$, $630$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:
 * $\map {\sigma_0} {714} = 16$, $\map \phi {714} = 192$, $\map {\sigma_1} {714} = 1728$

Also see

 * Integers whose Ratio between Divisor Sum and Phi is Square