1820
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Number
$1820$ (one thousand, eight hundred and twenty) is:
- $2^2 \times 5 \times 7 \times 13$
- The $13$th pentatope number after $1$, $5$, $15$, $35$, $70$, $126$, $210$, $330$, $495$, $715$, $1001$, $1365$:
- $1820 = \ds \sum_{k \mathop = 1}^{13} \dfrac {k \paren {k + 1} \paren {k + 2} } 6 = \dfrac {13 \paren {13 + 1} \paren {13 + 2} \paren {13 + 3} } {24}$
- The $35$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $925$, $1001$, $1080$, $1162$, $1247$, $1335$, $1426$, $1520$, $1617$, $1717$:
- $1820 = \ds \sum_{k \mathop = 1}^{35} \paren {3 k - 2} = \dfrac {35 \paren {3 \times 35 - 1} } 2$
- The $69$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $1335$, $1365$, $1426$, $1457$, $1520$, $1552$, $1617$, $1650$, $1717$, $1751$:
- $1820 = \ds \sum_{k \mathop = 1}^{35} \paren {3 k - 2} = \dfrac {35 \paren {3 \times 35 - 1} } 2$