Highly composite numbers are those positive integers with more divisors than any smaller number. In other words, they are the record setting numbers for the divisor function. The sequence begins 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ... .

Now, the divisor function τ(n) counts the number of divisors of n, including 1 and itself. If n has a prime factorization that is the product of p_i^a_i, with i running from 1 to k, then τ(n)=(a₁+1)(a₂+1)...(a_k+1).

We can generalize this function by replacing 1 in this product by a complex z.

We could say τ_z(n) = (a₁+z)(a₂+z)...(a_k+z).

Then we can define highly z-composite numbers to be those positive integers for which |τ_z(n) | is greater than |τ_z(j)| for all j < n.

Also, for each n, we can consider the set z of complex numbers for which n is highly z-composite. For most n, there will be no such z. But for some, this set might be quite interesting.

I will have to draw some pictures.

Some time ago, I wrote this bit about a proof I had wanted to see for a long time, and my disappointment when I saw "it".

I don't know if that inspired them or not, but today I discovered that two fellows added this sequence to the On-Line Encyclopedia of Integer Sequences, with a link to my page.

I thought it amusing.

I also discovered that googling "worst proof ever" brings up my page as the first hit. Ha!

I added two more sound files to my page of integer sequence noise. The new sequences are the palindromic numbers (like 121, 32423, etc.) and the set of values taken on by sigma, the sum of divisors function (1,3,4,6,7,8,12,13,14, etc.). The first doesn't sound very interesting, as palindromic numbers are a bit too rare and regular, but the second one is quite noisy. There is a bit of a peak at 44100/6=7350 hz, since multiples of 6 tend to be values of the sum of divisors function, but it's pretty uniformly noisy.

I finally got around to creating a page with my noise from integer sequences.

Put your headphones on and check it out and let me know what you think (or hear), and if you have suggestions for other sequences.