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Approximately Count the Number of Arithmetical Number that are Perfect Squares

Given is a Collection of Prime Numbers $p_i$ which we denote as $P$.
For every Prime $p_i \in P$ we now have another Collection $Q_i$, which contains all the numbers we can make from $\{1, \dots, p_i\}$ and are also Perfect Squares. We also have a Collection $\mathcal{N}_i$ which counts the number of such Arithmetical Numbers (I’ll give the definition below), that is an element of $Q_i$, for each Prime $p_i \in P$.
Now, I want to know, how is it possible to formulate an algorithm that tells me approximately how many numbers belong to $Q_i$, for each $p_i \in P$.
The only thing I know at the moment is that the size of $\mathcal{N}_i$ is of order $O(log_{p_i}\sqrt{p_i})$.
In general, we cannot approximate the size of any Collection in a better way, right?

A:

Well, since you say a Prime $p$ is used $n$ times, let’s call it a factor of $n$. As the case you want is the case where $n$ is even, we may assume that for primes $p_i$ this is $2m$ for some
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