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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.

Remark 19.1.2.

Incidentally, very (very) often one will see \(\sigma_0(n)\) written as \(\tau(n)\text{,}\) sometimes also as \(d(n)\text{.}\) Usually \(\sigma_1(n)\) is written simply \(\sigma(n)\text{,}\) though Euler apparently used \(\int n\) in his writings (can you think why?).
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?
  • A formula, at least for some input types.
  • See if at least a limited form of multiplicativity (recall Definition 18.1.2) holds.
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.