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:
- $33 \, 550 \, 336 = \ds \sum_{k \mathop = 1}^{8191} k = \dfrac {8191 \times \paren {8191 + 1} } 2$
Also see
- Previous ... Next: Perfect Number
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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.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $33,550,336$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $33,550,336$