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

Let $a,b,c$ be positive integers. It is known that there are infinitely many positive integers not representated by $a{x}^2+b{y}^2+c{z}^2$ with $x,y,z\in\mathbb Z$\mathrm{.}Incontrast,weconjecturethatanynaturalnumberisrepresentedby$\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor +\lfloor z^2/c\rfloor$with$x,y,z\in\mathbb Z$if$(a,b,c)\not=(1,1,1),(2,2,2)$, and that any natural number is represented by $\lfloor T_x/a\rfloor+\lfloor T_y/b\rfloor+\lfloor T_z/c\rfloor$with$x,y,z\in\mathbb Z$,where$T_x$ denotes the triangular number $x(x+1)\mathrm{/}2$. We confirm this general conjecture in some special cases; in particular, we prove that $$\{x^2+y^2+\lfloor \frac{z^2}{5}\rfloor :x,y,z\in \mathbb{Z}and2\nmid y\}=\{1,2,3,\dots \}$$ and $$\{\lfloor \frac{x^2}{m}\rfloor +\lfloor \frac{y^2}{m}\rfloor +\lfloor \frac{z^2}{m}\rfloor :x,y,z\in \mathbb{Z}\}=\{0,1,2,\dots \}form=5,6,15.$$ We also pose several conjectures for further research; for example, we conjecture that any integer can be written as $x^4-y^3+z^2$, where $x$, $y$ and $z$ are positive integers.