AdamJTP

Today, @ViolaMaths introduced me to Beethoven numbers.

They’re not mathematically interesting like other special numbers. They’re just pandigital numbers (without any zeros) in base 10.

Now, it can be seen that the digits sum to a 45, which is divisible by 9.  This means all Beethoven numbers are divisible by 9.

This means that if we to factorize Beethoven numbers we’d see a lot of 3s.

Anyway here are factorizations of the first few Beethoven numbers:

123456789=3 3 3607 3803
123456798=2 3 3 3 3 769 991
123456879=3 3 3 3 3 3 7 13 1861

@ViolaMaths’s Beethoven number factorized is:

396457812=2 2 3 3 37 297641

The largest prime we’d see is 109739359 in:

987654231= 3 x 3 x 109739359

Graphing the frequency

So here’s the frequency of the prime factors for the first few primes:

Looking at all the data and unsurprisingly lower primes are more frequent.

And that graph stretches a LONG way to the right.

Let’s look at that line on the left:

We can see from the mini-table above that there’s a spike at 3 (as expected). I’d guessed that we’d see a number of 2s a bit lower than 9! (362880), the actual number was 323145.

There’s a bit of dip in the 5s, but as we’ve taken out the numbers ending in zero (Beethoven didn’t write a zeroth symphony).

If Beethoven had only written 7 symphonies then we wouldn’t get any 3s.

You’d expect to see effects like that when the numbers have a relationship to the number base.

Conclusion

None really. It allowed me to make some guesses about the frequency before running the test app, and the graph looked pretty much as I expected it to.

A fun exercise – nothing more interesting than that.