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Consider the following Diophantine equation $$x^2 + xy + y^2 = n\,.$$ For a particular positive integer $n$, the number of solutions $\left(x,y\right)$ such that $x$ and $y$ are integers is given by the function $S(n)$.

The function $S(n)$ is not one-to-one. For example, each number $n$ in the set $$\{1, 4, 9, 16, 25, 36\}$$ corresponds with $S(n) = 6$.

In increasing order, starting from $n = 1$, the first $n$ such that $S(n) = 36$ is $637$.

What is the $500$th $n$ such that $S(n) = 36$?

Any precise approach or hint from where I should start the problem?

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    $\begingroup$ Where does this problem come from? $\endgroup$
    – saulspatz
    Jul 7, 2020 at 14:45
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    $\begingroup$ @EDX surely the RHS would be n+xy no? I hope I'm not making a fool of myself $\endgroup$
    – wesupportthepalace
    Jul 7, 2020 at 14:46
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    $\begingroup$ Since $x^2+xy+y^2=\left(x+\frac{y}{2}\right)^2+3\left(\frac{y}{2}\right)^2$ we need to consider ring $\mathbb{Z}[\omega]$, where $\omega=\frac{1}{2}+\frac{\sqrt{3}}{2}i$. This numbers are called Eisenstein integers, if I'm not mistaken and your question is how to find number of solutions of the equation $N(z)=n$. $\endgroup$
    – richrow
    Jul 7, 2020 at 14:54
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    $\begingroup$ The solutions are well understood, you should start there: The set of integers $n$ expressible with $n=x^2+xy+y^2$ $\endgroup$
    – Sil
    Jul 7, 2020 at 14:55
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    $\begingroup$ Actually $S(637)=36$, while $S(636)=0$. $\endgroup$
    – Robert Israel
    Jul 7, 2020 at 16:07

1 Answer 1

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The function $S$ corresponds to OEIS sequence A004016. From there you can get several implementations, for example in PARI:

 {a(n) = if( n<1, n==0, 6 * sumdiv( n, d, (d%3==1) - (d%3==2)))}; /* Michael Somos, May 20 2005 */
 k=0; for (n=1, oo, if (a(n)==36, print (k++ " " n); if (k==500, break)))
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