Note that , the number of divisors of . Thus is simply the number of divisors of .
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
There will be products formed by taking one number from each sum, which is the number of divisors of . Clearly all possible products are divisors of . Furthermore, all of those products are unique since each positive integer has a unique prime factorization.
Since all of these products are added together, we can conclude this gives us the sum of the divisors.
Sum of kth Powers of Divisors
Inspired by the example of the sum of divisors, we can easily see that the sum of the powers of the divisors is given by where are the distinct prime divisors of .
This is proven in a very similar way to the case.
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