* Definition.* The * sum of
divisors* function is given by

The * number of divisors* function is given by

* Example.* Recall that a number is * perfect* if it's equal to the sum of its divisors
other than itself. It follows that a number n is perfect if .

* Example.*

I want to show that and are multiplicative. I can do most of the work in the following lemma.

* Lemma.* The divisor sum of a multiplicative
function is multiplicative.

* Proof.* Suppose f is multiplicative, and let
be the divisor sum of f. Suppose . Then

Then

Now , so if and , then . Therefore, multiplicativity of f implies

Now every divisor d of can be written as , where and . Going the other way, if and then . So I may set , where , and replace the double sum with a single sum:

This proves that is multiplicative.

* Example.* The identity function is multiplicative: for *all* m, n (so *a fortiori* for
). Therefore, the divisor sum of is multiplicative. But

Hence, the sum of divisors function is multiplicative.

* Example.* The constant function is multiplicative: for *all* m, n (so *a fortiori* for
). Therefore, the divisor sum of I is
multiplicative. But

Hence, the number of divisors function is multiplicative.

I'll use multiplicativity to obtain formulas for and in terms of their prime factorizations (as I did with ). First, I'll get the formulas in the case where n is a power of a prime.

* Lemma.* Let p be prime. Then:

* Proof.* The divisors of are 1, p, , ..., . So the sum of the divisors is

And since the divisors of are 1, p, , ..., , there are of them, and

* Theorem.* Let , where the p's are distinct
primes and for all i. Then:

* Proof.* These results follow from the
preceding lemma, the fact that and are multiplicative, and the fact that the prime power
factors are pairwise relatively prime.

Here is a graph of for .

Note that if p is prime, . This gives the point , which lies on the line . This is the line that you see bounding the dots below.

Here is a graph of for .

If p is prime, . Thus, repeatedly returns to the horizontal line , which you can see bounding the dots below.

* Example.*
, so

* Example.* For each n, there are only finitely
many numbers k whose divisors sum to n: . For k divides itself, so

This says that k must be less than n. So if I'm looking for numbers whose divisors sum to n, I only need to look at numbers less than n. For example, if I want to find all numbers whose divisors sum to 42, I only need to look at .

Copyright 2008 by Bruce Ikenaga