Divisor function
The divisor function is denoted and is defined as the sum of the
th powers of the divisors of
. Thus
where the
are the divisors of
.
Counting divisors
Note that $\sigma_0(n) = d_1^0 + d_2^0 + \ldots + d_r^0 = 1 + 1 + \ldots + 1 = r</math>, the number of divisors of $n$. Thus is simply the number of divisors of
.
Example
Consider the task of counting the divisors of 72.
- First, we find the prime factorization of 72:
- 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
divisors.
We can now generalize. Let the prime factorization of be
. Any divisor of
must be of the form
where the
are integers such that
for
. Thus, the number of divisors of
is
.
Sum of divisors
The sum of the divisors, or , is given by
![$\sigma_1(n) = (1 + p_1 + p_1^2 +\cdots p_1^{e_1})(1 + p_2 + p_2^2 + \cdots + p_2^{e_2}) \cdots (1 + p_n + p_n^2 + \cdots + p_n^{e_n}).$](http://latex.artofproblemsolving.com/a/0/1/a01fdafba55f143dafb4631b35b41fea26a857b2.png)