Sum of divisor powers

The function σ_k takes an integer n and returns the sum of the kth powers of divisors of n. For example, the divisors of 14 are 1, 2, 4, 7, and 14. If we set k = 3 we get

σ₃(n) = 1³ + 2³ + 4³ + 7³ + 14³ = 3096.

A couple special cases may use different notation.

• σ₀(n) is the number of divisors n and is usually denoted d(n), as in the previous post.
• σ₁(n) is the sum of the divisors of n and the function is usually written σ(n) with no subscript.

In Python you can compute σ_k(n) using divisor_sigma from SymPy. You can get a list of the divisors of n using the function divisors, so the bit of code below illustrates that divisor_sigma computes what it’s supposed to compute.

    n, k = 365, 4
    a = divisor_sigma(n, k)
    b = sum(d**k for d in divisors(n))
    assert(a == b)

The Wikipedia article on σ_k gives graphs for k = 1, 2, and 3 and these graphs imply that σ_k gets smoother as k increases. Here is a similar graph to those in the article.

The plots definitely get smoother as k increases, but the plots are not on the same vertical scale. In order to make the plots more comparable, let’s look at the kth root of σ_k(n). This amounts to taking the Lebesgue k norm of the divisors of n.

Now that the curves are on a more similar scale, let’s plot them all on a single plot rather than in three subplots.

If we leave out k = 1 and add k = 4, we get a similar plot.

The plot for k = 2 that looked smooth compared to k = 1 now looks rough compared to k = 3 and 4.