Questions tagged [floor-function]

The floor function, also known as the greatest integer function, maps a real number $x$ to the greatest integer less than or equal to $x$ (often denoted $\lfloor x \rfloor$). See also (ceiling-function).

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What is the mathematical notation for rounding a given number to the nearest integer?

What is the mathematical notation for rounding a given number to the nearest integer? So like a mix between the floor and the ceiling function.
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Prove that $\lceil 2x\rceil =\lceil x\rceil +\lceil x+1/2\rceil$ -1 for all $x$ in $\mathbb R$ [on hold]

I don't know how to approach the problem. I have searched for different ceiling and floor properties but none of them seemed to help. Or simply I just don't know how to tackle the question.
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Limit of a greatest integer function (sided limit) [closed]

What is the value of $\lim\limits_{x\to 0^+} \dfrac{b}{x}\left\lfloor\dfrac{x}{a}\right\rfloor$ for $a>0$ and $b>0$. Note that $\lfloor x\rfloor$ denotes the greatest integer less than or equal ...
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Prove that for $n\in\mathbb{N}$, $\sum_{k=1}^{\infty}\left[\frac{n}{5^{k}}\right]=15\iff\left[\frac{n}{5}\right]=13.$

How to show that the following relation? : for $n\in\mathbb{N}$, $$\sum_{k=1}^{\infty}\left[\frac{n}{5^{k}}\right]=15\iff\left[\frac{n}{5}\right]=13.$$ It's not obvious to me. Can anyone help me? ...
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Surjectivity of floor of harmonic sequence

Define $$H_n := \displaystyle\sum_{k=1}^n \dfrac 1k$$ The problem asks to prove that the map $\phi:\mathbb{N}^\star \longrightarrow \mathbb{N}^\star$ defined by $$\phi(n) := \lfloor H_n \rfloor$$...
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Floor function simplification identities

I can't seem to find any identity(if any)for division/multiplication involving floor functions: for example $$\lfloor{\frac{n-1}{2}}\rfloor\cdot 2$$ I know does not simplify down to $$n-1$$.
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Calculate the sum with floor function.

Let $a$ be a positive number. Calculate the sum $$\sum_{1\le n\le x}\left\lfloor \sqrt{n^{2}+a} \right\rfloor$$ I tried to calculate first $\left\lfloor \sqrt{n^{2}+a} \right\rfloor-n$. But probably ...
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Computing $\sum_{k=1}^n (a^k \bmod m)$

I would like to find a closed form solution for $$\sum_{k=1}^n (a^k \bmod m)$$ $$0<a<m, n > 0$$ Note that the mod operator is within the brackets. If a closed form solution does not exist, ...
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How to calculate sum of floor functions.

Let $f(x)$ is real valued function. How to calulate or at least find lower and upper bound of sum: $$\sum_{1\le n \le x}\left\lfloor f(n) \right\rfloor$$? For example when $f(n)=\frac{x}{n}$ ...
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Computing the integral $\int \limits_{1}^{\infty}\left(\frac{1}{\lfloor{x}\rfloor}-\frac{1}{x}\right)$ …

Prove that $$\large\int \limits_{1}^{\infty}\Bigg(\dfrac{1}{\lfloor{x}\rfloor}-\dfrac{1}{x}\Bigg)dx=\lim \limits_{n \to \infty} \Bigg(-\ln(n) + \sum \limits_{k=1}^n\dfrac{1}{k}\Bigg)$$ I was reading ...
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What exactly is $\lfloor -0.5 \rceil$?

Suppose $g(x) = \lfloor x \rceil$ converts a real number $x$ into its nearest integer. I know $g(0.4) = 0$ $g(0.6) = 1$ But what are those? $g(0.5) = ?$ $g(-0.5) = ?$
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Is $1 + \lceil{\log_{10}x}\rceil = \lceil{1 + \log_{10}x}\rceil$ can anyone please prove or disprove it. [closed]

it seems to be true, but I dont have the proof
Integral $\int_{0}^{n^{2}} \lfloor \sqrt{t} \rfloor \rm dt$
To find the integral $\int_{0}^{n^{2}} \lfloor \sqrt{t} \rfloor \rm dt$ if $n \geq 1$ is an integer and $\lfloor \cdot \rfloor$ is the floor function, I began with the observation that \int_{0}^{n^{...
For which $n$ is $\frac{n!}{4}$ equal to $\left\lfloor \sqrt{\frac{n!}{4}}\right\rfloor\left(\left\lfloor\sqrt{\frac{n!}{4}}\right\rfloor + 1\right)$?
For how many values of $n$, is $\frac{n!}{4}$ equal to $\left\lfloor \sqrt{\frac{n!}{4}}\right\rfloor\left(\left\lfloor\sqrt{\frac{n!}{4}}\right\rfloor + 1\right)$? Further more, is there a way to ...