1540

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Number

$1540$ (one thousand, five hundred and forty) is:

$2^2 \times 5 \times 7 \times 11$

The $4$th after $1$, $10$, $120$ of the $5$ tetrahedral numbers which are also triangular.

The $20$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $56$, $84$, $120$, $165$, $220$, $286$, $364$, $455$, $560$, $680$, $816$, $969$, $1140$, $1330$:
$1540 = \ds \sum_{k \mathop = 1}^{20} \frac {k \paren {k + 1} } 2 = \dfrac {20 \paren {20 + 1} \paren {20 + 2} } 6$

The $28$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $703$, $780$, $861$, $946$, $1035$, $1225$, $1326$, $1431$:
$1540 = \ds \sum_{k \mathop = 1}^{28} \paren {4 k - 3} = 28 \paren {2 \times 28 - 1}$

The $55$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $990$, $1035$, $1081$, $1128$, $1176$, $1225$, $1275$, $1326$, $1378$, $1431$, $1485$:
$1540 = \ds \sum_{k \mathop = 1}^{55} k = \dfrac {55 \times \paren {55 + 1} } 2$