Listable sets

 Definition:

A set $A \subset \mathcal{Z}$ is listable if there is an algorithm that prints it.

Picked up this short and sweet definition from Undecidability in Number Theory

Clearly Diophantine sets are listable. Others include

• the set of all primes
• the set of Fibonacci numbers

A set of integers expressible as sum of three cubes is listable. 

For example, one can list elements using the following Mathematica code.

scp[z_] :=
 FindInstance[z == x^3 + y^3 + t^3, {x, y, t}, PositiveIntegers]
listS = Map[scp, Range[10]]

As we change the Range, we get larger and larger lists. The fact we have an algorithm that yields a list of elements is sufficient to characterize listable sets.