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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.

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  • $\begingroup$ this should be easy to look up. i know it's been done before $\endgroup$ Commented May 6, 2020 at 1:05
  • $\begingroup$ Standard name for the problem? $\endgroup$ Commented May 6, 2020 at 1:11
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    $\begingroup$ How about googling the title of this question? $\endgroup$ Commented May 6, 2020 at 2:22

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This problem is solved for $k\leq 5$ and open for $k\geq 6$. See Koukoulopoulos's paper for more details.

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