Section19.1Exploration: a new sequence of functions

Definition19.1.1
Let \(\sigma_k(n)\) be defined as the sum of the \(k\)th power of the (positive) divisors of \(n\), thus: \[\sigma_k(n)=\sum_{d\mid n}d^k\, .\]

Before doing any computing, think about what special information about a number \(\sigma_1\) and \(\sigma_0\) might encode.

Remark19.1.2

Incidentally, very (very) often one will see \(\sigma_0(n)\) written as \(\tau(n)\), sometimes also as \(d(n)\). Usually \(\sigma_1(n)\) is written simply \(\sigma(n)\), though Euler apparently used \(\int n\) in his writings.

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 \(n\). 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.

You might want to search for:

  • A formula, at least for some input types.
  • See if at least a limited form of multiplicativity 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 possible be bigger than \(n\) itself! So how big can these functions get, relative to \(n\)? How small?
  • Can anything be said about congruence values of these functions? (This is a little harder.)