Articles of floor function

$2 \lfloor x \rfloor \leq \lfloor 2x \rfloor \leq 2 \lfloor x \rfloor +1$

May I know the standard proof technique to prove such kind of inequalities. $2 \lfloor x \rfloor \leq \lfloor 2x \rfloor \leq 2 \lfloor x \rfloor +1$ Thanks!

how to sum the floors of ratios n/k when prime factorization of n is known

According to @harald-hanche-olsen the sum of the floors of ratios of $n/k$ is approximately: $$n(\ln n-1-\ln2)<\sum_{k=2}^{n-1}\Bigl\lfloor \frac nk\Bigr\rfloor<n\ln n.$$ If the prime factorization of $n$ is known (that is, if $n=\Pi_i{k_i^{\alpha_i}}$), can a closer approximation be obtained? In my case the exact value of $n$ is not important, just the accuracy of the sum of […]

Floor function to the base 2

I’m not a math guy, so I’m kinda confused about this. I have a program that needs to calculate the floor base $2$ number of a float. Let say a number $4$, that base $2$ floor would be $4$. Other examples : $5 \to 4$ $6 \to 4$ $8 \to 8$ $12 \to 8$ $16 […]

Floor Function Bound?

I am trying to prove or disprove the following bound: $2+\left(n-\left\lfloor\dfrac n k\right\rfloor k\right)\ge \left\lfloor\dfrac n k\right\rfloor$, where $2<k\le \left\lfloor\dfrac {n-1} 2\right\rfloor$, $n>4$, $k, n$ are coprime, and $n,k\in\mathbb N$. Any suggestions, solutions, or hints would be appreciated!

If $x – \lvert x \rvert + \frac{1}{x} – \lvert \frac{1}{x} \rvert = 1$ then $x$ is irrational

This question already has an answer here: Proving that $x$ is irrational if $x-\lfloor x \rfloor + \frac1x – \left\lfloor \frac1x \right\rfloor = 1$ 2 answers

How to find the values that make this interval enclose an integer

Suppose I have an interval that looks like this: $\left[\frac{k}{\lfloor m \rfloor }, \frac{k}{\lfloor mr \rfloor}\right)$ $m$ and $r$ are positive real numbers, but are constants in this problem. Here is the question: Which integer values of $k$ make this interval include at least one integer? I have tried various ways to tackle this problem, […]

Partial sum of Beatty sequence of $e$

Is there a fast algorithm to compute the sum of first $n$ terms of the Beatty sequence $e$? That is, I want to compute $$\lfloor e\rfloor+\lfloor 2e\rfloor+\lfloor 3e\rfloor+\lfloor 4e\rfloor+\lfloor 5e\rfloor+\cdots+\lfloor ne\rfloor$$ for very large $n$. Here $n$ can have up to $4000$ digits. OEIS:

Solving an equation with floor function

I need some help regarding this question. Solve the following equation in natural number $x$, where $m,k$ are fixed naturals: $m\left\lfloor \sqrt{\dfrac{x}{k}}\right\rfloor = x$. I think the answer is $x = m\left\lfloor\dfrac{m}{k}\right\rfloor$. But I do not know how to prove this claim. I would be happy if somebody would help me solve this problem. Thank […]

How do I solve floor function integral equation?

This equations comes from my other question, and I thought it was ok to create another question about the same exercise. So I have to solve the equation: $$\int_0^{\lfloor x\rfloor}\lfloor t\rfloor^2\mathrm dt+\lfloor x\rfloor^2(x-\lfloor x\rfloor)=2(x-1),$$which is the same as $$\frac{(\lfloor x\rfloor^2-\lfloor x\rfloor)(2\lfloor x\rfloor-1)}{6}+\lfloor x\rfloor^2(x-\lfloor x\rfloor)=2(x-1),$$ here’s the graph of $f(x)=\int_0^{\lfloor x\rfloor}\lfloor t\rfloor^2\mathrm dt+\lfloor x\rfloor^2(x-\lfloor x\rfloor)$: One […]

For $n \in \mathbb{N}$ $\lfloor{\sqrt{n} + \sqrt{n+1}\rfloor} = \lfloor{\sqrt{4n+2}\rfloor}$

This is Exercise 3.20 from Apostol’s book. Many of them seem tough and here is one of them which I am struggling with. For $n \in \mathbb{N}$, prove that this identity is true: $$\Bigl\lfloor{\sqrt{n} + \sqrt{n+1}\Bigl\rfloor} = \Bigl\lfloor{\sqrt{4n+2}\Bigl\rfloor}$$