Sum of all divisors
A number-theoretic property
About Sum of all divisors
The sum of all positive divisors of n is often denoted σ(n). It appears in the definition of perfect numbers (σ(n) − n = n), abundant numbers (σ(n) − n > n), and deficient numbers (σ(n) − n < n). Here we mean the sum including n itself; the sum of proper divisors is σ(n) − n.
Key features
- For a prime p, σ(p) = 1 + p.
- Perfect numbers satisfy σ(n) = 2n (sum of divisors equals twice the number).
- Multiplicative: if gcd(a, b) = 1 then σ(ab) = σ(a)σ(b).
Examples: