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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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1answer
31 views

Prove $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{n}+\sqrt{n+2}]$ for all integers $n\ge 1$

This is the full question Prove that of the two equations $$ [\sqrt{n}+\sqrt{n+1}] = [\sqrt{n}+\sqrt{n+2}] \\ [\sqrt[3]{n}+\sqrt[3]{n+1}] = [\sqrt[3]{n}+\sqrt[3]{n+2}]$$ the first one holds for ...
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1answer
49 views

Prove that $\sum_{j=1}^{\infty} \left[ \frac{n}{2^j}+\frac{1}{2} \right] = n$ for interger $n\ge 1$

This is the full question: Consider an integer $n\ge 1$ and the integers $i$,$1\le i \le n$. For each $k=0,1,2,\cdots$, find the number of $i$'s that are divisble by $2^k$ but not by $2^{k+1}$.Thus ...
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For what $n\in \mathbb{N}$ is $[nx]+[ny] \ge [x]+[y]+[(n-1)(x+y)] \, \forall x,y \in \mathbb{R}$?

For what $n\in \mathbb{N}$ is $[nx]+[ny] \ge [x]+[y]+[(n-1)(x+y)] \,\, \forall x,y \in \mathbb{R}$ ? Where [x] denotes the greatest integer function/ floor function(denoted by $\lfloor x\rfloor$) of $...
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Solve for equations involving floor function [duplicate]

Solve for real $x$: $$\frac{1}{\lfloor x \rfloor} + \frac{1}{\lfloor 2x \rfloor} = x - \lfloor x \rfloor + \frac{1}{3}$$ Hello! I hope everybody is doing well. I was not able to solve the above ...
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1answer
48 views

The proof of $(n+1)!(n+2)!$ divides $(2n+2)!$ for any positive integer $n$

Does $(n+1)!(n+2)!$ divide $(2n+2)!$ for any positive integer $n$? I tried to prove this when I was trying to prove the fact that ${P_n}^4$ divides $P_{2n}$ where $n$ is a positive integer, where $P_{...
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Proving Infinite Limit from Definition, Floor Function

I want to formally prove that the function $$f(x)=\textrm{floor}\left(\frac{1}{\pi}\left(x-\frac{\pi}{2}\right)\right)$$ tends to infinity as $x\rightarrow\infty$. More specifically, I am having ...
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2answers
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Floor function of a non-integer Cauchy sequence also a Cauchy sequence?

Let $a_n$ be a Cauchy sequence such that $a_n$ converges to a non-integer value. Then if we have a sequence $b_n$ defined as $b_n \leq a_n < b_n+1$, will $b_n$ be a Cauchy sequence? Intuitively, I ...
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Prove that for $\alpha\not\in\mathbb{Q},\alpha>0$, $([n+n\alpha])_{n\in\mathbb{N}}\sqcup([n+n\alpha^{-1}])_{n\in\mathbb{N}}=\mathbb{N} $

My Quesion: Prove that for $\alpha\not\in\mathbb{Q},\alpha>0$, $([n+n\alpha])_{n\in\mathbb{N}}\bigcup([n+n\alpha^{-1}])_{n\in\mathbb{N}}=\mathbb{N}$, where $[k]$ means the integral part of $k\in\...
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$n^\text{th}$ term of $1,2,\dots,4,4,5,5,\dots,9,9,9,10,10,10,\dots,16,16,16,16,17,17,17,17,\dots$ [closed]

Consider the sequence $$1,2,3,4,4,5,5,6,6,7,7,8,8,9,9,9,10,10,10,11,11,11,12,12,12,13,13,13,14,14,14,15,15,15,16,16,16,16,\dots$$ Which is $$\underset{\left \lfloor \sqrt{1} \right \rfloor \text{ ...
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Floor function error in Desmos?

I ran over a problem with desmos after playing around with floor functions and want to know what is my thinking error or if the problem lies with Desmos. In the Screenshot you can see the function ...
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Piecewise Relation for the Floor and Ceiling Functions

I am trying to write the piecewise function notation for the floor and ceiling step functions. This is my own rule - I am just looking for confirmation that it is valid. For $f(x) = [x]$ and $c(x) = ...
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Find the thousandth number in the sequence of numbers relatively prime to $105$.

Suppose that all positive integers which are relatively prime to $105$ are arranged into a increasing sequence: $a_1 , a_2 , a_3, . . . .$ Evaluate $a_{1000}$ By inclusion exclusion principle I ...
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1answer
28 views

Prove or disprove if x∈Z, then $x=⌈\frac{x}{2}⌉+⌊\frac{x}{2}⌋$

Prove or disprove if x∈Z $$x=⌈\frac{x}{2}⌉+⌊\frac{x}{2}⌋$$ I'm just unsure how to go about this question and can't find decent examples in my textbook. I'm assuming that the assumption is true based ...
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1answer
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Help me with an interesting number theory proof: Find all different [n/k] - floor function

A positive integer $n$ is given. Let's consider all numbers of the form $\lfloor\frac{n}{k}\rfloor$ where $k$ is a positive integer. Prove that there are no more than $2\sqrt n + 1$ different numbers ...
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solving a equation (floor function)

I am trying to solve the following problem: For what real numbers x is: ⌊2x⌋=4⌊x⌋+3? I'm not sure how to deal with the floor functions, so I have no idea where to start. If someone could walk me ...
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Integral of $[\frac{1}{x}]^{-1}$ on $(0,1]$

I found in a book this exercise: Prove that $\int_0^1 \frac{1}{[{\frac{1}{x}}]}dx=+\infty$ In my proof i find that this integral is finite. Here is my proof: By Beppo-Levi: $\int_0^1 \frac{1}{[...
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If $x\Big\lfloor x \big\lfloor x\lfloor x \rfloor \big\rfloor \Big\rfloor=2018$, find $x$

If $x\Big\lfloor x \big\lfloor x\lfloor x \rfloor \big\rfloor \Big\rfloor=2018$, find $x$. My working: if $x$ is positive then by estimation it must be in $(6,7)$ and for this interval I : $x\Big\...
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Understanding a particular algorithm that computes $\sum\limits_{i=1}^n\lfloor\frac ni\rfloor$ with complexity less than $O(n)$ [duplicate]

Is there a better way to get the summation of $$\sum\limits_{i=1}^n\left\lfloor\frac ni\right\rfloor$$ I would like to know a way that is better than the complexity of $O(n)$ ... because there seems ...
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The Airy functions and miscellaneous sequences of odd integers without repeated prime factors

I'm curious about the following miscellaneous conjectures, for which I hope that one can to get a counterexample. I add the encyclopedia Wikipedia's article for the Airy functions $\operatorname{Ai}(...
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1answer
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To find a sum of i(floor2019/i-floor2019/(i-1), i range from 2 to 2019

$$ \sum_{\text{i}=2}^{2019}{\text{i}\left( \left\lfloor \frac{\text{n}}{\text{i}}\right \rfloor -\left\lfloor \frac{\text{n}}{\text{i}-1} \right\rfloor \right)} $$ where n=2019 $$$$Obviously, this is ...
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Largest and smallest number that has k less digits in next number base

Let $n,k\in\mathbb N$, and $n\ge 2,k\ge 0$. Let $a_k(n)$ be the largest number that has $k$ less digits in number base $n+1$ than in base $n$. Let $b_k(n)$ be the smallest number that has $k$...
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1answer
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On the sum of sum of divisors $\sum_{a=1}^{N} D \left({\left\lfloor{\frac{N}{a}}\right\rfloor}\right)$.

where $D \left({x}\right)$ is the sum of divisors. This sum comes from my work on the number of reducible monic cubics. This is a two part question. By writing out all the divisors $\tau \left({a}\...
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Partial Divisor Summatory Function Calculation

I'm aware that $D(n)$ can be calculated in O(sqrt(n)) time. Can $ D(n, k) $ also be calculated in O(sqrt(n)) time? What's the best algorithm? For example, if $n = 8$ and $k = 3$, then $ D(n, k) = \...
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Prove that $\lceil 2x\rceil =\lceil x\rceil +\lceil x+1/2\rceil$ -1 for all $x$ in $\mathbb R$ [closed]

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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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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5answers
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Proper way to solve $\lim\limits_{x\rightarrow\infty}\left[\frac{6}{x}\right]\frac{x}{3}$

I made a problem for my friends, but not everyone agreed with my answer. $$\lim_{x\rightarrow\infty}\left[\frac{6}{x}\right]\frac{x}{3}+\lim_{x\rightarrow\infty}\frac{6}{x}\left[\frac{x}{3}\right]+\...
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Showing that $\lfloor\frac{x-1}3\rfloor=\lfloor\frac{x}3+\frac23\rfloor-1$ and $\lfloor\frac{x+1}3\rfloor=\lfloor\frac{x}3+\frac13\rfloor$ [duplicate]

I have 2 questions about the floor functions: 1) $\left\lfloor \frac{x-1}{3}\right\rfloor =\left\lfloor \frac{x}{3}+\frac{2}{3}\right\rfloor -1$ 2) $\left\lfloor \frac{x+1}{3}\right\rfloor =\left\...
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2answers
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Showing that $\lfloor\frac{x-1}3\rfloor=\lfloor\frac{x}3+\frac23\rfloor-1$ and $\lfloor\frac{x+1}3\rfloor=\lfloor\frac{x}3+\frac13\rfloor$.

I have 2 questions about the floor functions: 1) $\left\lfloor \frac{x-1}{3}\right\rfloor =\left\lfloor \frac{x}{3}+\frac{2}{3}\right\rfloor -1$ 2) $\left\lfloor \frac{x+1}{3}\right\rfloor =\left\...
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Calculate $ \left \lfloor \frac{2017^{3}}{2015 \cdot 2016} - \frac{2015^{3}}{2016 \cdot 2017} \right \rfloor $

Calculate $$ \left \lfloor \frac{2017^{3}}{2015 \cdot 2016} - \frac{2015^{3}}{2016 \cdot 2017} \right \rfloor $$ attempt: $$ \frac{2017^{3}}{2015 \cdot 2016} - \frac{2015^{3}}{2016 \cdot 2017} = \...
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3answers
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Find the positive solutions for $x + 2 \{ x \} = 3 \lfloor x \rfloor$

Find the positive solutions for $x + 2 \{ x \} = 3 \lfloor x \rfloor$ attempt: Notice that the equation can be rewritten as $$ x + 2 \{ x\} = 2 \lfloor x \rfloor + x - \{x\}$$ $$ 3 \{x\} = 2 \...
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1answer
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Proof that with $b < 0$, $a$ mod $b \in (b,0]$

I was trying to prove that $a$ mod $b \in (b,0]$ when $b < 0$. To do that basically I need to prove that $a - b\lfloor a/b \rfloor < 0$ which means that we need to prove that $b\lfloor a / b\...
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2answers
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Let x ∈ R. Show that $2\lfloor x \rfloor \leq \lfloor 2x \rfloor \leq 2\lfloor x \rfloor + 1 $

I already checked some questions about this statement, however I can't understand why the first inequality is true. We know that $\lfloor x \rfloor \leq x$, then $2\lfloor x \rfloor \leq 2x$. In ...
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2answers
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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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1answer
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How to prove $ \lfloor\log{(n+1)} / 2 \rfloor+1 = \lfloor\log{(n+1)}\rfloor$

I was trying to prove the equation below using the floor definition but finally I have given up. I have no idea how to prove it. Could anyone give me a hint how to start? $ \lfloor\log{(n+1)} / 2 \...
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1answer
125 views

The maximum of real function with 4 prime parameters and $\lfloor \ \rfloor$

Let $a$,$b$,$c$ and $d$ be prime numbers such that $a>b>c>d$. Let $x$ be an integer greater than $a$. Let $f(x) = \left(\dfrac{x}{a}\right) – \left(\left(\dfrac{x}{ab}\right) + \left(\dfrac{...
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0answers
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Given $a_{n+1}=a_{n}+\frac{1}{a_{n}}$, how can I find $\lfloor a_{1000} \rfloor$? [duplicate]

Given $a_{n+1}=a_{n}+\frac{1}{a_{n}}$, and $a_0=5,$ how can I find $\lfloor a_{1000} \rfloor$? I've tried coming up with reasonable bounds, but to no avail (ex. ones like $\sqrt{x^2+1} < x + \frac{...
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1answer
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The units digit of $1!+2!+3!+4!!+5!!+\dots+k\underset{\left \lfloor \sqrt{k} \right \rfloor \text{ times}}{\underbrace{!!!\dots!}}$

For natural numbers $n\ge m$, let $n\underset{m \text{ times}}{\underbrace{!!!\dots!}}=n(n-m)(n-2m)(n-3m)\dots$ where all factors are natural numbers (we exclude $0$ and negative factors). Question: ...
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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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1answer
42 views

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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3answers
56 views

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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2answers
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Can I apply the floor function to the left- and right hand side this way?

I'm wondering if I can apply the floor function to both sides like below? This isn't all I want to do I just want to know if the operation is legal. Thanks! $$ a + x < b + y \to \lfloor a + x \...
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1answer
33 views

Range of $f(x) = \frac{1}{1+\lfloor x \rfloor}$

I need to compute the range of: $$f(x) = \frac{1}{1+\lfloor x \rfloor}$$ Intuitively, the range is $\frac{1}{n}$, where $n \in \Bbb Z^*$, but I want to prove that this is the range of the function ...
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1answer
58 views

If you take any natural number greater than three, take the square root and round it off …

If you take any natural number greater than three, take the square root and round it off, from the result you take the square root and round it off again, and so on. Show that ,at some point, this ...
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1answer
39 views

Evaluating $f(n)$ using mod function and floor function

For $n =1,2,3,\dots,12$, we are given that $$\begin{bmatrix} n & f(n)\\ 1 & 0\\ 2 & 3\\ 3 & 2\\ 4 & 5\\ 5 & 0\\ 6 & 3\\ 7 & 5\\ 8 & 1\\ 9 & 4\\ 10 &...
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3answers
80 views

Is this the correct way to integrate this: $\int_0^5 (x^2+1) d \lfloor x \rfloor$ , where $\lfloor\cdot \rfloor$ is the greatest integer function?

This Question was asked by my teacher and the solution he presented is this: $$\int_0^5 (x^2+1) d \lfloor x \rfloor.$$ Before integrating the above, lets see this: $\int_a^b f'(x)g(x) + g'(x)f(x)dx ...
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3answers
58 views

Prove $⌊2x⌋+⌊2y⌋≥⌊x⌋+⌊y⌋+⌊x+y⌋$

Prove the following for all real $x$ i. $⌊2x⌋+⌊2y⌋≥⌊x⌋+⌊y⌋+⌊x+y⌋$ ii. $⌊x⌋-2⌊x/2⌋$ is equal to either $0$ or $1$ For ($ii$.) I attempted to split it into cases of whether the fraction part {$x$} is ...
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1answer
47 views

Evaluating series sum with floor

Is there a closed form expression or a good approximation for the following expression: \begin{equation} \sum^{\infty}_{k=n+1} {\left\lfloor \frac{k}{n+1} \right\rfloor p^k} \end{equation} Knowing ...
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1answer
44 views

$\sum_{k=1}^{\infty }\left \lfloor \frac{n}{p^k} \right \rfloor=\frac{n-S_n}{p-1}$

If $p$ is a prime number, $n$ is a natural number, and $S_n$ is the sum of the digits of $n$ when expressed in base $p$. Prove that $\sum_{k=1}^{\infty }\left \lfloor \frac{n}{p^k} \right \rfloor=\...
3
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2answers
88 views

Roots of $(x-\lfloor x\rfloor)^2+(x-\lfloor x\rfloor)\left\lfloor{1\over x -\lfloor x\rfloor}\right\rfloor=1$

Can you help me to find -some analytical- roots of the following function ? I know $\sqrt2$ is a root and I think there are infinitely many roots,according to plot provided by WolframAlpha. $$ (x-\...
3
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1answer
80 views

The digital root of a tower of exponents, $d(\underset{\text{The number of }2 \text{'s is }2013}{\underbrace{2^{2^{2^{.^{.^{.^{2}}}}}}}})$

For a natural number $n$, the digital root of $n$ is the value obtained by an iterative process of summing digits. The digital root of $n$ is denoted by $d(n)$. Examples; $d(142)=7$, $d(123785)=8$ ...