This project is the second in week long series about some topics that my son’s math club will be studying at school. The topic for today is sums of divisors of a positive integer.

I was a little surprised to see this topic on the list because this is a pretty advanced topic. In fact, in just the short time that we spent talking about the problem this morning we arrived at an unsolved problem! Fun project, though, and we’ll talk more about this topic tomorrow.

I started by having the kids play around on Mathematica to see what they would notice about divisor sums. Some of the things they found:

(1) 9240 has the highest divisor sum out of all the integers from 1 to 10,000,
(2) the divisor sum for primes is one more than the prime, and
(3) the divisor sum for perfect squares is always odd.

Next we spent a little time looking at the divisor sum for the perfect squares. It was interesting to hear why they thought the divisor sum would always be odd:

After the discussion in front of the computer, we went to the white board to look more carefully at the sum of the divisors for perfect squares. We started off this discussion by looking at 36. We also look at 225 to see what happenes in an example where all of the factors are odd:

Now we tried to tackle the general case – why does all perfect squares have an odd number of odd factors?

After that discussion, we returned to the computer to explore when the sum of the divisors of a prime number?

Here’s the start of that discussion:

We finished up this part of the discussion by looking to see how far the pattern we found in the first part would continue. We’ll learn a bit more about this pattern tomorrow:

Finally we looked quickly at the super abundant numbers. These are numbers with unusually large divisor sums. In particular, when you take the sum of the divisors and divide by the number, you get a ratio which is larger than any number below it. Here’s a quick introduction, though my attempt to write a quick program on the fly fails so I had to break this part into two pieces:

So, a fun conversation with the kids today. Excited to dig more into sums of factors tomorrow.