On the class numbers of indefinite binary quadratic forms with discriminant $dp^2$

A number of results on the average values of the class numbers of indefinite binary quadratic forms with discriminants divisible by a large square are proved. The main result is as follows. Let $d=4n^2+1$. Then
$$ \mathop{{\sum}'}_{1\le n\le X}\frac1{h(d)}\sum_{2X\le p\le3X}h(dp^2)=O(X^2), $$
where $h(d)$ is the class number for the discriminant $d$ and $\sum'$ means that the summation is performed over the square-free $d$ only.