# Talking about “sums of divisors” with kids and also a pi surprise

I didn’t do a very good job managing the time on this project today. The trouble is that there are lots of different directions to go with the ideas and we walked down a lot of different paths.

But – I think this is a great topic to show off the beauty of math and we end with an amazing connection between sums of divisors of integers and $\pi$.

The topic of sums of divisors of an integer came up in my younger son’s weekend enrichment math program yesterday. I thought it would make for a good topic for a project, so I gave it a go this morning.

The first part of the project was mostly about divisors and the kinds of questions that we could ask about them. A lot of the discussion here is about a question you can ask about the product of a number’s divisors:

Next we began to look at the sum of the divisors of a few different numbers. The boys noticed a few patterns – including a pattern in the powers of 2.

At the end we were looking to see if we could find patterns in the powers of 3.

It was proving to be a little difficult to find the pattern in the powers of 3, but we kept trying. After few ideas that didn’t quite help us write down the pattern, they boys had an idea that got us there.

At the end of this video I showed them that the sum of the divisors of powers of 6 was connected with the sum of the divisors of powers of 2 and powers of 3.

To wrap up I wanted to show some larger patterns in divisor sums, so we moved to Mathematica to play around a bit.

While I was doing the same playing around last night I accidentally stumbled on an amazing fact: As n gets large, the average of the sum of the divisors of the numbers from 1 to n is approximately $(\pi^2 / 12)*n$.

Number theory sure has some fun surprises 🙂

This is definitely a fun topic and also one that could be used in a variety of ways (arithmetic review, intro to number theory, computer math, . . . ). I wish that I’d presented it better. Probably it needs more than one project to really fit in all of the ideas, though.