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Let $a,b,c$ be positive integers. It is known that there are infinitely manypositive integers not representated by $ax^2+by^2+cz^2$ with $x,y,z\in\mathbbZ$. In contrast, we conjecture that any natural number is represented by$\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor +\lfloor z^2/c\rfloor$ with$x,y,z\in\mathbb Z$ if $(a,b,c)\not=(1,1,1),(2,2,2)$, and that any naturalnumber is represented by $\lfloor T_x/a\rfloor+\lfloor T_y/b\rfloor+\lfloorT_z/c\rfloor$ with $x,y,z\in\mathbb Z$, where $T_x$ denotes the triangularnumber $x(x+1)/2$. We confirm this general conjecture in some special cases; inparticular, we prove that$$\left\{x^2+y^2+\left\lfloor\frac{z^2}5\right\rfloor:\ x,y,z\in\mathbb Z\\mbox{and}\ 2\nmid y\right\}=\{1,2,3,\ldots\}$$ and$$\left\{\left\lfloor\frac{x^2}m\right\rfloor+\left\lfloor\frac{y^2}m\right\rfloor+\left\lfloor\frac{z^2}m\right\rfloor:\x,y,z\in\mathbb Z\right\} =\{0,1,2,\ldots\}\ \ \mbox{for}\ m=5,6,15.$$ We alsopose several conjectures for further research; for example, we conjecture thatany integer can be written as $x^4-y^3+z^2$, where $x$, $y$ and $z$ arepositive integers.

Natural numbers represented by $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$