The divisor function sigma_k(n) is defined as the sum of the kth powers of positive divisors of integer n.
- sigma_0(14) = 1^0 + 2^0 + 7^0 + 14^0 = 1 + 1 + 1 + 1 = 4
- sigma_1(30) = 1^1 + 2^1 + 3^1 + 5^1 + 6^1 + 10^1 + 15^1 + 30^1 = 72
- sigma_3(6) = 1^3 + 2^3 + 3^3 + 6^3 = 1 + 8 + 27 + 216 = 252
- If k = 0, that is sigma_0(n), then the function returns the number of divisors of n.
- If k = 1, that is sigma_1(n), then the function returns the sum of all divisors of n.
- For a prime number p,
- sigma_0(p) = 1^0 + p^0 = 1 + 1 = 2
- sigma_1(p) = 1^1 + p^1 = p + 1
- sigma_0(p^n) = 1^0 + (p^1)^0 + (p^2)^0 + ... + (p^n)^0 = 1 + n * 1 = n + 1