715
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Number
$715$ (seven hundred and fifteen) is:
- $5 \times 11 \times 13$
- The $2$nd of the $5$th (and largest known) pair of consecutive integers whose product is a primorial:
- $714 \times 715 = 510 \, 510 = 17 \#$
- The $10$th pentatope number after $1$, $5$, $15$, $35$, $70$, $126$, $210$, $330$, $495$:
- $715 = \ds \sum_{k \mathop = 1}^{10} \dfrac {k \paren {k + 1} \paren {k + 2} } 6 = \dfrac {10 \paren {10 + 1} \paren {10 + 2} \paren {10 + 3} } {24}$
- The $22$nd pentagonal number after $1$, $5$, $12$, $22$, $35$, $51$, $70$, $92$, $117$, $145$, $176$, $210$, $247$, $287$, $330$, $330$, $376$, $425$, $477$, $532$, $590$, $651$:
- $715 = \ds \sum_{k \mathop = 1}^{22} \paren {3 k - 2} = \dfrac {22 \paren {3 \times 22 - 1} } 2$
- The $43$rd generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $392$, $425$, $442$, $477$, $495$, $532$, $551$, $590$, $610$, $651$, $672$:
- $715 = \ds \sum_{k \mathop = 1}^{22} \paren {3 k - 2} = \dfrac {22 \paren {3 \times 22 - 1} } 2$