Skip to main content

Sum of divisor powers

The function σk 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 σk(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 σk gives graphs for k = 1, 2, and 3 and these graphs imply that σk 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 σk(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.

The post Sum of divisor powers first appeared on John D. Cook.



from John D. Cook https://ift.tt/2QEhFy2
via IFTTT
Labels:

Comments

Post a Comment

Popular posts from this blog

Controlling legend appearance in ggplot2 with override.aes

[This article was first published on Very statisticious on Very statisticious , and kindly contributed to R-bloggers ]. (You can report issue about the content on this page here ) Want to share your content on R-bloggers? click here if you have a blog, or here if you don't. In ggplot2 , aesthetics and their scale_*() functions change both the plot appearance and the plot legend appearance simultaneously. The override.aes argument in guide_legend() allows the user to change only the legend appearance without affecting the rest of the plot. This is useful for making the legend more readable or for creating certain types of combined legends. In this post I’ll first introduce override.aes with a basic example and then go through three additional plotting scenarios to how other instances where override.aes comes in handy. Table of Contents R packages Introducing override.aes Adding a guides() layer Using the guide argument in scale_*() Changing multiple aesthetic par...
Post a Comment
Read more

Zooming in on maps with sf and ggplot2

(This article was first published on r-bloggers – WZB Data Science Blog , and kindly contributed to R-bloggers) When working with geo-spatial data in R, I usually use the sf package for manipulating spatial data as Simple Features objects and ggplot2 with geom_sf for visualizing these data. One thing that comes up regularly is “zooming in” on a certain region of interest, i.e. displaying a certain map detail. There are several ways to do so. Three common options are: selecting only certain areas of interest from the spatial dataset (e.g. only certain countries / continent(s) / etc.) cropping the geometries in the spatial dataset using sf_crop() restricting the display window via coord_sf() I will show the advantages and disadvantages of these options and especially focus on how to zoom in on a certain point of interest at a specific “zoom level”. We will see how to calculate the coordinates of the display window or “bounding box” around this zoom point. Loading t...
Post a Comment
Read more