Properties of greatest integer function

Question

I am curious to know some properties of the floor functions, for instance, $\lfloor a \cdot x \rfloor$, $\lfloor a1\cdot x1+a2\cdot x2 \rfloor$, etc. Is there any book that contains such properties ?

Answer 1

• There is Legendre's formula which counts the number of positive integers less than or equal to a number $n$ which are not divisible by any of the first $k$ primes: $$\begin{align} &\phi(n,k)=\lfloor n \rfloor-\sum_{p_i\le k}\left\lfloor \dfrac{ n }{(p_i)}\right\rfloor+\sum_{p_i<p_j\le k}\left\lfloor\dfrac{ n}{(p_ip_j)}\right\rfloor-\sum_{p_i<p_j<p_m\le k}\left\lfloor \dfrac{n}{(p_ip_jp_m)}\right\rfloor+\dots \end{align}$$

which tells us that total number of times a prime $p$ divides $n!$ is $$\sum_{k=1}^{\infty}\lfloor\frac{n}{p^k}\rfloor$$

• For positive integers $\lfloor \sqrt{n}+\sqrt{n+1}\rfloor=\lfloor\sqrt{4n+2}\rfloor$
• $\lfloor2x\rfloor+\lfloor2y\rfloor\ge \lfloor x \rfloor+\lfloor y \rfloor+\lfloor x+y \rfloor$
• $\lfloor \frac n2 \rfloor- \lfloor \frac{-n}{2} \rfloor=n$ for integers $n \ge 0$.

You should look up in books on Discrete mathematics or combinatorics. Also see wikipedia link