Most formally, a
perfect number is a
natural number n for which σ(n) = 2n. The σ function (lowercase sigma, for those without σ in their browser's character set) returns the sum of the
aliquot parts of a number, aka the sum of all the number's divisors (
all divisors, not just the
proper ones). If σ(n) < 2n, then n is called
deficient. If σ(n) > 2n, then n is called
abundant.
There has been some interest lately in finding "multiple perfect numbers," numbers for which σ(n) = kn, for integers k greater than 2. The human race has yet to discover a multiple perfect number with k = 11, although many have been discovered for k = 10.