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perfect numbers

Definition/Summary
A perfect number is a number which is the sum of its proper divisors (half the sum of its total divisors).  Even perfect numbers are a Mersenne prime times a power of two; odd perfect numbers are not known to exist.

Equations
Sum-of-divisors function: $$\sigma(n)=\sum_{k|n}k$$ $$\sigma(p^aq^b)=\sigma(p^a)\sigma(q^b)\;\;(p,q\text{ relatively prime})$$ $$\sigma(p^a)=\frac{p^{a+1}-1}{p-1}$$  Definition of N perfect: $$2N=\sigma(N)$$  Form of an even perfect number: $$N=M_p(M_p+1)/2=2^{p-1}(2^p-1)$$ where M_p is a Mersenne prime.

Recent forum threads on perfect numbers

Breakdown
Mathematics 					> Number Theory 					>> Sequences

See Also

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Extended explanation
The first two perfect numbers are:  6 = 1 + 2 + 3 = $2^{2-1} (2^2-1)$  28 = 1 + 2 + 4 + 7 + 14 = $2^{3-1} (2^3-1)$  The next two are:  496 = $2^{5-1} (2^5-1)$  8128 = $2^{7-1} (2^7-1)$

Commentary

seyalbert @ 01:16 PM Apr10-11
???????

seyalbert @ 01:16 PM Apr10-11
Someone knows a formula for add perfect numbers.
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