The well-known problem is acquiring a cardinality of the set of distinct numbers in the multiplication table n x m.

The very problem has been discussed in-depth and, as such, I require no further input on it by itself. There has been, however, a significant amount of debate about it on StackOverflow, namely this question:

http://stackoverflow.com/questions/24614798/find-the-number-of-distinct-numbers-in-multiplication-table

and this question/bounty:

http://stackoverflow.com/questions/24714104/how-to-solve-erdos-uniques-multiples-in-about-an-iterations/24851735#24851735

While, as far as I understand, the problem has currently only O(n^2) computational solutions (strictly speaking, k*n^2 iterations, with k=0.5), while the asymptotic size of the set is equal to
$$\left|\lbrace a\cdot b:\ a,b\leq N\rbrace\right|\asymp \frac{N^2}{(\log N)^c(\log\log N)^{3/2}}$$ where $$c=1-\frac{(1+\log \log 2)}{\log 2}.$$
(Ford, 2008).

Still, there has been significant amount of dispute about it by certain individuals, *convinced* there is an O(n) solution to the problem [calculating A(n)], and that they have found it.

As far as my knowledge goes, there is no explicit way to generate a set of size A(n) and to calculate it's cardinality without at least A(n) operations. Also, there currently exists no solution to acquiring the exact value of A(n) without generating the set and counting its unique elements. Since the reference solutions, as the one send by David are, as far as I understand, O(n^2), I'm asking for some clarification on this matter [having the problem computationally solved strictly below O(n^2)] from people with knowledge broader than mine.

related:

http://mathoverflow.net/questions/151270/how-many-different-numbers-can-be-obtained-as-product-of-first-n-natural-numbe

http://mathoverflow.net/questions/31663/distinct-numbers-in-multiplication-table

http://mathoverflow.net/questions/108912/number-of-elements-in-the-set-1-cdots-n-times-1-cdots-n?lq=1