1001

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Number

$1001$ (one thousand and one) is:

$7 \times 11 \times 13$

The $26$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $477$, $532$, $590$, $651$, $715$, $782$, $852$, $925$:
$1001 = \displaystyle \sum_{k \mathop = 1}^{26} \paren {3 k - 2} = \dfrac {26 \paren {3 \times 26 - 1} } 2$

The $51$st generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $610$, $651$, $672$, $715$, $737$, $782$, $805$, $852$, $876$, $925$, $950$:
$1001 = \displaystyle \sum_{k \mathop = 1}^{26} \paren {3 k - 2} = \dfrac {26 \paren {3 \times 26 - 1} } 2$

The $11$th pentatope number after $1$, $5$, $15$, $35$, $70$, $126$, $210$, $330$, $495$, $715$:
$1001 = \displaystyle \sum_{k \mathop = 1}^{11} \dfrac {k \paren {k + 1} \paren {k + 2} } 6 = \dfrac {11 \paren {11 + 1} \paren {11 + 2} \paren {11 + 3} } {24}$

The $4$th pentagonal number after $1$, $5$, $22$ which is also palindromic:
$1001 = \displaystyle \sum_{k \mathop = 1}^{26} \paren {3 k - 2} = \dfrac {26 \paren {3 \times 26 - 1} } 2$

The $7$th positive integer after $1$, $2$, $7$, $11$, $101$, $111$ whose cube is palindromic:
$1001^3 = 1 \, 003 \, 003 \, 001$

Historical Note

The most important cultural significance of the number $1001$ occurs in the collection of medieval Arabic folk tales known as One Thousand and One Nights.

The specific significance of the number $1001$ derives from the poetically rhetorical device: a surprisingly large number ($1000$) and then some (and $1$).