The number of divisors of $n^2 = 11^{4022} \cdot 2011^{22}$ is:
$$N = (4022 + 1)(22 + 1) = 92529.$$
We can pair the divisors of $n^2$ in a way $d$ and $\frac{n^2}{d}$ where one divisor is greater than $n$, another one is less. The number $n$ does have a pair, therefore the number of divisors that are less than $n$ is $\frac{92529-1}{2} = 46264$.

The number $N_1$ of divisors of $n$ that are less than $n$ is:
$$N_1 = (2011+1)(11+1)-1 = 24143.$$
So the number we are looking for is $46264 - 24143 = 22121$.