As odd squares are congruent to $1$ modulo $8$, any integer of the form $4^k(8m+7)$ with $k,m\in\mathbb N=\{0,1,2,\ldots\}$ cannot be written as the sum of three squares.
To avoid such congruence obstacles in representation problems, I suggest a variant of squares by using Euler's totient function $\varphi$. It is easy to see that all the numbers $$\varphi(n^2)=n\varphi(n)\ \ (n\in\mathbb Z^+=\{1,2,3,\ldots\})$$ are pairwise distinct.
QUESTION: Can we write each $n\in\mathbb Z^+$ as $x^2+y^2+\varphi(z^2)$ with $x,y\in\mathbb N$ and $z\in\mathbb Z^+$?
Actually, on Oct. 1, 2015, I even made the following stronger conjecture.
Conjecture. Any integer $n>1$ can be written as $x^2+y^2+\varphi(z^2)$, where $x,y\in\mathbb N$, $x\le y$, $z\in\mathbb Z^+$, and $y$ or $z$ is prime.
For the number of ways to write $n$ in this way, see http://oeis.org/A262311. For example, $13$ has a unique required representation: $13=1^2+2^2+\varphi(4^2)$ with $2$ prime, and $94415$ has a unique required representation: $$94415=115^2+178^2+\varphi(223^2)\ \ \text{with}\ 223\ \text{prime}.$$ I have verified the conjecture for all $n=2,\ldots,10^6$.
Though the conjecture might be quite challenging, the weaker version in the question should be on the research level. Any comments are welcome!
