It is a basic but extremely useful fact that the collection of prime powers can be characterized in the following way:

There is a field with $q$ elements if and only if $q$ is a prime power. Furthermore if it exists then this field is unique up to isomorphism.

This simple result has been exploited using the theory of finite fields to prove many interesting results, which we will not list here. My question is,

Is there an analogous characterization for positive integers which are products of two distinct prime powers?

To avoid triviality let us say that the characterization should be presented independently of the prime factorization of the number involved. So we are looking for a statement like,

"There exists a set from a certain class which has property $P(n)$ if and only if $n$ is a product of two distinct prime powers,"

and both the set from the relevant class and the property $P(n)$ should be defined in a way that does not inherently depend on the prime factorization of $n$.