Consider multiplication operation $$f(x\_1,\dots, x\_k)=\prod\_{i=1}^kx\_i$$ where $x\_i\in\{1,\dots, n\_i\}$ with $n\_1,\dots, n\_k\in\{1,\dots,\infty\}$.


What is the cardinality of the range?


At $k =2$ with $n\_1=n\_2$ this is the standard Erdos multiplication table problem whose estimates are in [Distinct numbers in multiplication table](https://mathoverflow.net/questions/31663/distinct-numbers-in-multiplication-table/31666).



Generalized Erdős multiplication table problem
This problem is solved for $k\leq 5$ and open for $k\geq 6$. See [Koukoulopoulos's paper](https://arxiv.org/abs/1102.3236) for more details.


