NTIC Exploring a New Sequence of Functions

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:

$\begin{equation*} \sigma_k(n)=\sum_{d\mid n}d^k\text{.} \end{equation*}$

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Before doing any computing, think about what special information about a number

$\sigma_1$ and

$\sigma_0$ might encode.

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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?).

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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

$n\text{.}$

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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.

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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.