sumofdivisors

Sum of divisors, denoted σ(n), is the arithmetic function that assigns to a positive integer n the sum of all positive divisors of n, including 1 and n. It can be written as σ(n) = ∑_{d|n} d. The function is multiplicative: if gcd(a,b) = 1, then σ(ab) = σ(a)σ(b). For a prime power p^k, σ(p^k) = (p^{k+1} − 1)/(p − 1). Consequently, for n with prime factorization n = ∏ p_i^{a_i}, σ(n) = ∏ (p_i^{a_i+1} − 1)/(p_i − 1).

Examples illustrate its use: σ(6) = 1 + 2 + 3 + 6 = 12, and σ(28) = 1 + 2 + 4 + 7 + 14 + 28 = 56. A number n is called perfect if σ(n) = 2n; classic examples are 6 and 28, each equal to half of its divisor-sum. Numbers with σ(n) > 2n are abundant, while those with σ(n) < 2n are deficient.

σ is the Dirichlet convolution of the identity function id(n) = n with the constant function 1, i.e., σ = id * 1. It is studied for its growth and distribution properties; on average, the sum of σ(n) for n ≤ x satisfies ∑_{n≤x} σ(n) ∼ (π^2/12) x^2, so the average value of σ(n) among numbers up to x is about (π^2/12) x. Computationally, σ(n) can be obtained from the prime factorization of n, and sieve methods allow efficient computation of σ(n) for ranges of integers.

See also: Perfect number, Abundant number, Deficient number.