Divisor function

The divisor function is denoted $\sigma_k(n)$ and is defined as the sum of the $k$ th powers of the divisors of $n$ . Thus $\sigma_k(n) = \sum_{d|n}d^k = d_1 + d_2 + \cdots + d_r$ where the $d_i$ are the divisors of $n$ .

Counting divisors

Letting $k=0$ makes all of the terms in $d_1 + d_2 + \cdots + d_r$ equal to 1. Thus, The value of $\sigma_0(n)$ is simply the number of divisors of $n$ .

Example: 72

Consider the task of counting the divisors of 72.

First, we find the prime factorization of 72: $\displaystyle72=2^{3} \cdot 3^{2}.$

Since each divisor of 72 can have a power of 2, and since this power can be 0, 1, 2, or 3, we have 4 possibilities. Likewise, since each divisor can have a power of 3, and since this power can be 0, 1, or 2, we have 3 possibilities. By an elementary counting principle, we have $3\cdot 4=12$ divisors.

We can now generalize. Let the prime factorization of $n$ be $p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ . Any divisor of $n$ must be of the form $p_1^{f_1}p_2^{f_2} \cdots p_k^{e_k}$ where the $\displaystyle f_i$ are integers such that $0\le f_i \le e_i$ for $i = 1,2,\ldots, k$ . Thus, the number of divisors of $n$ is $(e_1+1)(e_2+1)\cdots (e_k+1)$ .