# 33,550,336

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## Number

$33 \, 550 \, 336$ is:

$2^{12} \times 8191$

The $5$th perfect number after $6$, $28$, $496$, $8128$:
$33 \, 550 \, 336 = \map {\sigma_1} {33 \, 550 \, 336} - 33 \, 550 \, 336 = 4096 \times 8191 = 2^{13 - 1} \paren {2^{13} - 1}$

The $4096$th hexagonal number:
$33 \, 550 \, 336 = \ds \sum_{k \mathop = 1}^{4096} \paren {4 k - 3} = 4096 \paren {2 \times 4096 - 1}$

The $8191$st triangular number:
$33 \, 550 \, 336 = \ds \sum_{k \mathop = 1}^{8191} k = \dfrac {8191 \times \paren {8191 + 1} } 2$

## Historical Note

The earliest record of the perfect nature of $33 \, 550 \, 336$ is in a medieval manuscript dating from $1456$.

When Hudalrichus Regius demonstrated in $1536$ that both $511 = 2^9 - 1$ and $2047 = 2^{11} - 1$ are composite, it was confirmed as being the $5$th perfect number.