I also got some interest in the problem of finding, a number between 1 and n^2, which has the highest number of factors (including 1 and itself). One can easily prove that such a number is not divisible by any prime number >= n, for n > 4. Try it.

Of course, this problem can be formalized as optimization problem.
Given that,

k1*log(2) + k2*log(3) + k3*log(5) + .... all prime logs with co-efficients < log(n)
(or, equivalently, 2^k1*3^k2*5^k3*7^k4*..... < n)

maximize (k1 + 1)*(k2 + 1)*(k3 + 1)*(k4 + 1)*....

Finally the number, with most factors is given by 2^k1*3^k2*5^k3*7^k4*.....