Uninteresting Numbers

There's a quasi-paradox about interesting numbers which says that all the natural numbers must be interesting. If they weren't, then there would be a smallest uninteresting number, which is a very interesting thing to be. So that's immediately a contradiction and we're forced to conclude that all the natural numbers are interesting. This is just a bit of fun with self-referential definitions, but it suggests an interesting exercise with trying to formally define "interesting".

We can start with some reasonable attributes that qualify a number as interesting, and use them to find the first handful of uninteresting numbers. For example, we might start by saying that 0 and 1 are interesting for being the additive and multiplicative identities (respectively), and that all primes are interesting. By this definition the first ten uninteresting numbers are:

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ...

Then we can look at the first few and try to find something interesting about them that we can add to our list of attributes. For example 4 is a square, or more generally it is n^k where k > 1. Removing those sorts of numbers gives us:

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ...

6 is 2×3, which makes it a semiprime. We may as well say semiprimes are interesting, which also takes out 10 and 14 and 15 and 21 and 22, leaving us with:

12, 18, 20, 24, 28, 30, 40, 42, 44, 45, ...

12 is 3×4, which makes it the product of at least two consecutive numbers, so that's mildly interesting as well. This also captures 20 and 24 and 30 and 42, so now we have:

18, 28, 40, 44, 45, 48, 50, 52, 54, 63, ...

18 is right between a pair of twin primes, and 28 is a perfect number, so let's grab both of those:

40, 44, 45, 48, 50, 52, 54, 63, 66, 68, ...

These numbers are getting rather boring, so let's stop there. But conceivably you could take this arbitrarily far. As long as the set of numbers with any given attribute have an asymptotic density of 0 (i.e., become arbitrarily rare as you go up), there will always be more uninteresting numbers to handle. And as long as the attributes you pick are efficiently computable, the computer should always be able to find them for you.