1001
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Number
$1001$ (one thousand and one) is:
- $7 \times 11 \times 13$
- The $4$th pentagonal number after $1$, $5$, $22$ which is also palindromic:
- $1001 = \ds \sum_{k \mathop = 1}^{26} \paren {3 k - 2} = \dfrac {26 \paren {3 \times 26 - 1} } 2$
- The $7$th positive integer after $1$, $2$, $7$, $11$, $101$, $111$ whose cube is palindromic:
- $1001^3 = 1 \, 003 \, 003 \, 001$
- The $11$th pentatope number after $1$, $5$, $15$, $35$, $70$, $126$, $210$, $330$, $495$, $715$:
- $1001 = \ds \sum_{k \mathop = 1}^{11} \dfrac {k \paren {k + 1} \paren {k + 2} } 6 = \dfrac {11 \paren {11 + 1} \paren {11 + 2} \paren {11 + 3} } {24}$
- The $12$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $22$, $101$, $111$, $121$, $202$, $212$ whose square is also palindromic integer
- $1001^2 = 1 \, 002 \, 001$
- The $26$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $477$, $532$, $590$, $651$, $715$, $782$, $852$, $925$:
- $1001 = \ds \sum_{k \mathop = 1}^{26} \paren {3 k - 2} = \dfrac {26 \paren {3 \times 26 - 1} } 2$
- The $51$st generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $610$, $651$, $672$, $715$, $737$, $782$, $805$, $852$, $876$, $925$, $950$:
- $1001 = \ds \sum_{k \mathop = 1}^{26} \paren {3 k - 2} = \dfrac {26 \paren {3 \times 26 - 1} } 2$
Also see
- Previous ... Next: Sequence of Palindromic Pentagonal Numbers
- Previous ... Next: Prime Factors of One More than Power of 10
- Previous ... Next: Sequence of Integers whose Cube is Palindromic
- Previous ... Next: Square of Small-Digit Palindromic Number is Palindromic
- Previous ... Next: Pentatope Number
- Previous ... Next: Pentagonal Number
- Previous ... Next: Generalized Pentagonal Number
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 $1$).
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1001$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1001$