1,533,776,805

Number

$1 \, 533 \, 776 \, 805$ is:

$3^2 \times 5 \times 11 \times 17 \times 19 \times 53 \times 181$

The $3$rd integer after $1$, $40 \, 755$ to be both pentagonal and hexagonal

The $5$th integer after $1$, $210$, $40 \, 755$, $7 \, 906 \, 276$ to be both pentagonal and triangular

$1 \, 533 \, 776 \, 805 = \ds \sum_{k \mathop = 1}^{27 \, 693} \paren {4 k - 3} = 27 \, 693 \paren {2 \times 27 \, 693 - 1}$

The $31 \, 977$th pentagonal number:

$1 \, 533 \, 776 \, 805 = \ds \sum_{k \mathop = 1}^{31 \, 977} \paren {3 k - 2} = \dfrac {31 \, 977 \paren {3 \times 31 \, 977 - 1} } 2$

The $55 \, 385$th triangular number:

$1 \, 533 \, 776 \, 805 = \ds \sum_{k \mathop = 1}^{55 \, 385} k = \dfrac {55 \, 385 \times \paren {55 \, 385 + 1} } 2$

$1 \, 533 \, 776 \, 805 = \ds \sum_{k \mathop = 1}^{31 \, 977} \paren {3 k - 2} = \dfrac {31 \, 977 \paren {3 \times 31 \, 977 - 1} } 2$