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 positive divisors of .
Contents
[hide]Counting divisors
Note that , the number of divisors of . Thus is simply the number of divisors of .
Example Problems
Demonstration
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 .
Introductory Problems
Sum of divisors
The sum of the divisors, or , is given by