It is anachronistic to say that to the Greeks a number was a member of the set of all integers greater than one. They had neither a formal nor a naive theory of sets. To us today the ideas of set theory seem intuitive and inevitable but until about 130 years ago the idea of completed infinity such as an infinite set was seen as very problematic, and it was not one of their categories of thought.
From the modern viewpoint, it is completely unnatural to distinguish between $1$ and the other natural numbers, as the Greeks did. Therefore this collection does not seem to appear naturally in modern mathematics. There is no standard notation for it because there is no need for it.