This is the multiplication table problem of Erdos. According to Kevin Ford, Integers with a divisor in
$(y,2y]$, Anatomy of integers, 65-80, CRM Proc. Lecture Notes, 46, Amer Math Soc 2008, MR 2009i:11113, the number of positive integers $n\le x$, which can be written as $n=m_1m_2$, with each
$m_i\le\sqrt x$, is bounded above and below by a constant times $x(\log x)^{-\delta}(\log\log x)^{-3/2}$, where $\delta=1-(1+\log\log2)/\log2$.

Erdos' work on this problem can be found (in Russian) in An asymptotic inequality in the theory of numbers, Vestnik Leningrad Univ. Mat. Mekh. i Astr. 13 (1960) 41-49.

Another reference is http://oeis.org/A027424 where a PARI program is given.