A young man wrote to me about asymptotics of the set of numbers represented integrally by $x^2 - n^3 y^2 $ when we assume that we know asymptotics for $x^2 - n y^2.$ I made several suggestions, of the same type as in my question Asymptotic for primitive sums of two squares , along with discussion of the class number (of indefinite binary quadratic forms) $h(4 n^3)$ as a multiple of $h(4n).$ However, I did not settle the manner, and I don't know analytic number theory.
The computer experiments I've been doing are inconclusive. For example, I've been finding the count of numbers from $-N$ to $N$ of $x^2 - 21 y^2,$ as opposed to $x^2 - 9261 y^2.$ For modest $N$ the ratio was about 3, which is the ratio of the class numbers. As I increased $N$ the ratio began increasing, also for primitive representations. Oh, the ratio of class numbers really is the ratio for represented primes.
Cute fact: the multiplier of the class number is tied to the Pell equation. Given $n,$ find the fundamental solution to $t^2 - n u^2 = 1.$ The multiplier is just $\gcd(n,u),$ as in $h(4 n^3) = \gcd(n,u) h(4n) \; . $ Who knew?
The question: given numbers integrally represented by $x^2 - n y^2,$ what proportion are represented by $x^2 - n^3 y^2 \; \; \; \;$ and does it make any difference if we insist on primitive representations?