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:
\(\ds 6\) | \(=\) | \(\ds 2^{2 - 1} \times \paren {2^2 - 1}\) | ||||||||||||
\(\ds 28\) | \(=\) | \(\ds 2^{3 - 1} \times \paren {2^3 - 1}\) | ||||||||||||
\(\ds 496\) | \(=\) | \(\ds 2^{5 - 1} \times \paren {2^5 - 1}\) | ||||||||||||
\(\ds 8128\) | \(=\) | \(\ds 2^{7 - 1} \times \paren {2^7 - 1}\) | ||||||||||||
\(\ds 33 \, 550 \, 336\) | \(=\) | \(\ds 2^{13 - 1} \times \paren {2^{13} - 1}\) | ||||||||||||
\(\ds 8 \, 589 \, 869 \, 056\) | \(=\) | \(\ds 2^{17 - 1} \times \paren {2^{17} - 1}\) |
Also see
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): sigma function
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): sigma function