It is known (sum of two squares theorem) that a number can be written as a sum of two squares (that is, as $n = x^2 + y^2$ for integers $x$ and $y$) if and only if, in its prime factorization, every prime congruent to $3$ modulo $4$ (namely, each prime $3$, $7$, $11$, $19$, $23$, $31$, etc.) occurs to an even power (possibly $0$).
It is further known (Landau–Ramanujan constant: 1, 2, 3) that the number of such numbers less than $x$ is asymptotically equivalent to $K \dfrac{x}{\sqrt{\ln x}}$ where $$K \approx 0.76422.$$
Now, what if we want to count only the numbers $n$ that can be written as $x^2 + y^2$ where both $x$ and $y$ are non-zero? This sequence is OEIS A000404 rather than OEIS A001481.
By the same reasoning that leads to the sum-of-two-squares theorem, these are the numbers $n$ such that in the prime factorization of $n$, every prime that is congruent to $3$ modulo $4$ occurs to an even power, and there is at least one another prime. (In other words, from the set of sums-of-two-squares, we're excluding only numbers of the form $m^2$ where every prime factor of $m$ is congruent to $3$ modulo $4$.)
Asymptotically, how many such numbers are there less than a given $x$? That is, I gather that it must be $K' \dfrac{x}{\sqrt{\ln x}}$ for some constant $K'$; what is the exact value of $K'$?
Some questions that look relevant, though I don't understand all of the mathematics involved: Numbers divisible only by primes of the form 4k+1 on MathOverflow, and, linked from it, Asymptotic for primitive sums of two squares on this site.
