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 $S$ ?