<s>[INST] If $n=2^{31}\cdot 3^{19}$ then how many divisors of $n^2$ are smaller than $n$ but they aren't divisor of $n$? If $n=2^{31}\cdot 3^{19}$ then how many divisors of $n^2$ are smaller than $n$ but they aren't divisor of $n$?
It is clear that the power of $2$ or $3$ should be bigger than how much there are in $n$.For example I found some cases:
Power of $2=32\Rightarrow$there are $19$ cases for the power of $3$
Power of $2=33\Rightarrow$there are $18$ cases for the power of $3$
Power of $2=34\Rightarrow$there are $18$ cases for the power of $3$
But it is hard to see how many numbers the power of $3$ or $2$ can be by increasing the power of the other.Any hints?
 [/INST] Notice that if $d|n^2$ and $d<n$ then $n^2/d>n$.
We conclude the number of divisors of $n^2$ less than $n$ are $\frac{\tau(n^2)-1}{2}$
So the answer is $\frac{\tau(n^2)-1}{2}-\tau(n)+1$
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