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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

 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 = $2^{7-1} (2^7-1)$