Section 19.1 Exploring a New Sequence of Functions
Definition 19.1.1.
For \(n>0\text{,}\) let \(\sigma_k(n)\) be defined as the sum of the \(k\)th power of the (positive) divisors of \(n\text{,}\) thus:
\begin{equation*}
\sigma_k(n)=\sum_{d\mid n}d^k\text{.}
\end{equation*}
Before doing any computing, think about what special information about a number \(\sigma_1\) and \(\sigma_0\) might encode.
Hopefully, you realized \(\sigma_1\) is adding all the divisors of \(n\) (including \(n\) itself), and that \(\sigma_0\) is the number of (positive) divisors of \(n\text{.}\)
Now, get ready to explore! Try to figure out as much as you can about these functions. If you’re in a group in a class, you can certainly save time by dividing up the initial computations among yourselves, then sharing that information so you have a bigger data set to look at.
Question 19.1.3.
Can you find some or all of the following for these functions?
You might also want to look at questions like these.
Can two different \(n\) yield the same \(\sigma_k\) (for a given \(k\))? If so, when – or when not? Can they be consecutive?
Is it possible to say anything about when one of these functions yields even results – or ones divisible by three, four, … ?
Clearly the size of these functions somehow is related to the size of \(n\) – for instance, it is obvious that \(\sigma_0(n)=\tau(n)\) can’t possibly be bigger than \(n\) itself! So how big can these functions get, relative to \(n\text{?}\) How small?
Can anything be said about congruence values of these functions? (This is a little harder.)
If you come up with a new idea, why not challenge someone else to prove it? See
Exercise Group 19.6.2–4 for past examples.