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σ(n) sums every divisor including n itself. Comparing the aliquot sum σ(n) − n with n classifies the number: equal is perfect, greater is abundant and smaller is deficient.
Sigma from factorisation
σ(p₁^e₁ ⋯ pₖ^eₖ) = Π (p^(e+1) − 1)/(p − 1)
Its proper divisors 1, 2, 4, 7 and 14 sum to exactly 28.
None is known, and if one exists it must exceed 10¹⁵⁰⁰. It is one of the oldest open problems in mathematics.