Shu wrote: Find all numbers $n$ that can be expressed in the form $n=k+2\lfloor\sqrt{k}\rfloor+2$ for some nonnegative integer $k$. All elements of set: $\{n | n \in N, n>4, n \not= m^2-1, n \not= m^2, m \in N \}$

Let $k = a^2 + b$ such that $a,b \in \mathbb{Z}^+_0$ and $b \le 2a$. Then $n = a^2 + b + 2a + 2 = (a+1)^2 + b + 1$. Since $0 \le b \le 2a$, $1 \le b+1 \le 2a+1$. Note that $(a+1)^2 + 1 \le n \le (a+1)^2 + 2a+1 = (a-2)^2 - 2$, thus $n$ cannot be one less from a square or the square itself. So $n$ can take all elements in the set $\{x : x \neq y^2-1, x \neq y^2, x,y \in \mathbb{Z}^+\} = \{2, 5, 6, 7, 10, 11, 12, 13, 14, \cdots \}$.