divisorsum

Divisor sum, also called the sum of divisors function, denoted σ(n), assigns to a positive integer n the sum of all positive divisors of n, including 1 and n itself. In symbols, σ(n) = ∑_{d|n} d. The function is defined for n ≥ 1.

If n has prime factorization n = ∏ p_i^{a_i}, then σ(n) = ∏ (p_i^{a_i+1}-1)/(p_i-1). This formula shows that σ is multiplicative: if gcd(m,n) = 1 then σ(mn) = σ(m) σ(n).

Example: σ(6) = 1+2+3+6 = 12.

Proper divisor sum s(n) = σ(n) - n excludes n itself. Numbers for which σ(n) = 2n are perfect numbers (6, 28, 496, ...). If σ(n) > 2n the number is abundant; if σ(n) < 2n it is deficient.

Besides its basic definition, σ(n) is used in various contexts: it appears in average order estimates (on average σ(n) ~ (π^2/6) n), and its Dirichlet generating function is ∑ σ(n) n^{-s} = ζ(s) ζ(s-1) for Re(s) > 2. The function is studied alongside the divisor-counting function τ(n), which counts the number of divisors of n.

Computationally, σ(n) can be computed efficiently from the prime factorization of n, and sieve methods allow rapid computation for many integers up to a limit.