40,755

Number

$40 \, 755$ (forty thousand, seven hundred and fifty-five) is:

$3 \times 5 \times 11 \times 13 \times 19$

The $2$nd integer after $1$ to be both pentagonal and hexagonal

The $3$rd integer after $1$, $210$ to be both pentagonal and triangular

$40 \, 755 = \ds \sum_{k \mathop = 1}^{143} \paren {4 k - 3} = 143 \paren {2 \times 143 - 1}$

$40 \, 755 = \ds \sum_{k \mathop = 1}^{165} \paren {3 k - 2} = \dfrac {165 \paren {3 \times 165 - 1} } 2$

$40 \, 755 = \ds \sum_{k \mathop = 1}^{285} k = \dfrac {285 \times \paren {285 + 1} } 2$

$40 \, 755 = \ds \sum_{k \mathop = 1}^{165} \paren {3 k - 2} = \dfrac {165 \paren {3 \times 165 - 1} } 2$