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Let $S$ be the set of positive integers $n$, such that the equation $$a^2+b^2+c^2=n$$ has a solution in non-negative integers but not in positive integers. Since $n\in S$ if and only if $4n\in S$, we can assume that $n$ is not divisble by $4$. Also, $n\equiv 3\mod 4$ can be ruled out because a number of the form $8k+3$ is always the sum of three positive squares and a number of the form $8k+7$ is not the sum of three squares.

The first positive integers in $S$ not being divisble by $4$, are :

$$[1, 2, 5, 10, 13, 25, 37, 58, 85, 130]$$

I did not find further such integers yet.

Is this list complete ? If not , how can I classify the numbers being in $S$ ?

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    $\begingroup$ Relevant OEIS entry? $\endgroup$ – Arthur Jun 11 '18 at 12:10
  • $\begingroup$ Isn't $9$ in $S$? $\endgroup$ – lhf Jun 11 '18 at 17:04
  • $\begingroup$ Also of course $n\equiv 3\bmod 4$ isn't always the sum of three positive squares (e.g. $n=7$); I presume the meaning there is that it's never the sum of one or two squares? $\endgroup$ – Steven Stadnicki Jun 11 '18 at 17:57
  • $\begingroup$ @StevenStadnicki Sorry, I got this wrong, I meant a number of the form $8k+3$. I edited the question $\endgroup$ – Peter Jun 11 '18 at 21:15
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    $\begingroup$ oeis.org/A051952 says this is an open problem. $\endgroup$ – lhf Jun 11 '18 at 21:49

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