Definition:Perfect Number/Definition 2

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Definition

A perfect number $n$ is a (strictly) positive integer such that:

$\map {\sigma_1} n= 2 n$

where $\sigma_1: \Z_{>0} \to \Z_{>0}$ is the divisor sum function.


Sequence of Perfect Numbers

The sequence of perfect numbers begins:

\(\ds 6\) \(=\) \(\ds 2^{2 - 1} \times \paren {2^2 - 1}\)
\(\ds 28\) \(=\) \(\ds 2^{3 - 1} \times \paren {2^3 - 1}\)
\(\ds 496\) \(=\) \(\ds 2^{5 - 1} \times \paren {2^5 - 1}\)
\(\ds 8128\) \(=\) \(\ds 2^{7 - 1} \times \paren {2^7 - 1}\)
\(\ds 33 \, 550 \, 336\) \(=\) \(\ds 2^{13 - 1} \times \paren {2^{13} - 1}\)
\(\ds 8 \, 589 \, 869 \, 056\) \(=\) \(\ds 2^{17 - 1} \times \paren {2^{17} - 1}\)


Examples

6

$6$ is a perfect number:

$1 + 2 + 3 = 6$


28

$28$ is a perfect number:

$1 + 2 + 4 + 7 + 14 = 28$


496

$496$ is a perfect number:

$1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496$


8128

$8128$ is a perfect number:

$1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128$


Also see

  • Results about perfect numbers can be found here.


Sources

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