Answer: 589

Small Hint:

Pair each divisor $d$ of $n^2$ with $\frac{n^2}{d}$

Big Hint:

Among the divisors below $n$, subtract those that already divide $n$

Solution:

The number $n^2=2^{62}3^{38}$ has $63\cdot39=2457$ divisors. Pairing $d$ with $\frac{n^2}{d}$ leaves only $n$ unpaired, so $\frac{2457-1}{2}=1228$ divisors lie below $n.$ The number $n$ has $32\cdot20=640$ divisors, of which $639$ are below $n.$ Therefore the requested count is $1228-639=589.$