Questions tagged [ceiling-and-floor-functions]

This tag is for questions involving the greatest integer function (or the floor function) and the least integer function (or the ceiling function).

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Size of set of quotient floors

Given natural numbers $n$ and $m$, what is the size of the set $$\left\{\left\lfloor \frac{n}{i} \right\rfloor : i\in\{1,\dots,m\} \right\}$$ For example, $n=10, m=3$ gives the set $\{3,5,10\}$ (size $...
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  • 111
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finding a constant $C>0$ for a sequence to satisfy a given inequality

Let $(u_n)$ be the sequence defined as follows $$u_0=1$$ $$u_n=u_{\left\lfloor \tfrac n 2 \right\rfloor} + u_{\left\lfloor \tfrac n 3 \right\rfloor} + u_{\left\lfloor \tfrac n 6 \right\rfloor}$$ ($n \...
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  • 45
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1 answer
80 views

Closed form solution of a sum involving the floor function

I am searching for a closed form solution for the following sum ($a \geq 0, b>0, c>0, n \geq 0$): $$ S(a,b,c,n) = \sum_{k=0}^{n} \left\lfloor \frac{a+k\,b}{c} \right\rfloor $$ I already found a ...
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1 vote
1 answer
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Find a smooth function such that $f(m)=\lceil\frac{m}{3}\rceil, m\in\Bbb{Z}$

I've created a small challenge for myself, stated below; Find $f_1(x),f_2(x)$, and $f_3(x)$ if $f_n(x)$ is a smooth function s.t. $f_n(m)=\lceil\frac{m}{n}\rceil\space\space \forall \space\space m\in\...
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2 votes
3 answers
73 views

Calculate the sum of $\left\lfloor \sqrt{k} \right\rfloor$

I'm trying to calculate for $n\in \mathbb{N}$ the following sum : $\sum_{k=1}^{n^2}\left\lfloor \sqrt{k} \right\rfloor$. I tried putting in the first terms, which gave me $\sum_{k=1}^{n^2}\left\lfloor ...
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-1 votes
1 answer
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What are simplification rules for expressions involving floor division?

I encountered something weird while programming in Python. I understand that (31//16)*4is 4 because 31//16 evaluates to 1 and <...
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0 votes
1 answer
36 views

Proof of floor function is monotone

Can anyone give a hint to approach this question? I have no idea so far, $f(x_1) = \lfloor x_1 \rfloor = n_1$ where ; $n_1 \leq x_1 < n_1 + 1$ $f(x_2) = \lfloor x_2 \rfloor = n_2$ where ; $n_2 \leq ...
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  • 99
11 votes
2 answers
252 views

Limit of $n^2$ and a recurrence relation with ceiling function

For all positive integer $n$ we define a finite sequence in the following way: $n_0 = n$, then $n_1\geq n_0$ and has the property that $n_1$ is a multiple of $n_0-1$ such that the difference $n_1 - ...
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  • 535
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How to solve equations involving ceiling?

how can an equation involving ceilings be solved? (c is a known constant, looking for n)?: $$c=n-60\left \lceil \frac{n}{650} \right \rceil-70\left \lceil \frac{n}{600} \right \rceil-25\left \lceil \...
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1 answer
55 views

Help solving a floor equation

I have given the following equation, with the additional info that $x$ and $c$ are positive integers and $p$ is a decimal with $0 \le p \le 1$. My goal is to solve for x: $x = \lfloor xp + c\rfloor$ I'...
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1 vote
1 answer
56 views

How to calculate limit $\lim_{x\to\infty}\left(\lfloor x\rfloor+\frac{1}{2}\right)\ln x-\ln(\lfloor x\rfloor!)$

While uses plot on www.wolframalpha.com, i found a problem: The function $$f(x)=\left(\lfloor x\rfloor+\frac{1}{2}\right)\ln x-\ln(\lfloor x\rfloor!)$$ approximate the function $$g(x)=x-\frac{1}{2}\ln\...
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  • 1,339
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1 answer
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Geometric series sum confusion

Suppose I have a value k whose value is $\log_2n$ and sum of geometric series $1+2+4+8+...+2^k$ is $2^{k + 1} -1$ . Now I am calculating the complexity of a simple nested loop program and the above ...
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  • 75
-1 votes
3 answers
115 views

Trouble finding the floor function of a given expression.

I am currently learning a bit about number theory - currently studying continuous fractions and came up with the following task: Task. Show that the floor function of $$ \frac{n + \sqrt{n^2+4}}{2},$$ ...
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  • 667
9 votes
5 answers
382 views

$\left\lfloor\frac{8n+13}{25}\right\rfloor-\left\lfloor\frac{n-12-\left\lfloor\frac{n-17}{25}\right\rfloor}{3}\right\rfloor$ is independent of $n$

If $n$ is a positive integer, prove that $$\left\lfloor\frac{8n+13}{25}\right\rfloor-\left\lfloor\frac{n-12-\left\lfloor\frac{n-17}{25}\right\rfloor}{3}\right\rfloor$$ is independent of $n$. Taking $$...
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  • 5,858
2 votes
0 answers
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Calculate the sum of ceils of lg

Calculate $f(n)=\sum_{k=1}^n\lceil \lg k\rceil$ when $n>1$. I can understand the solution from textbook, but I cant figure out why my solution gets wrong answer. Hope someone can point out my ...
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1 vote
0 answers
25 views

Is this epsilon-delta proof involving the floor function correct?

I'm reading the book "Calculus a Rigorous First Course" by Daniel J Velleman. Currently stuck on exercise 11 from section 2.3 which is the following. Use the given formula for $\delta$ to ...
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0 answers
91 views

Showing $ \left \lceil \frac{a}{b} \right \rceil \leq \frac{a+b-1}{b}$ [duplicate]

I'm trying to convince myself how one can hypothesize that $$ \left \lceil \frac{a}{b} \right \rceil \leq \frac{a+b-1}{b}$$ or alternatively that $$ \left \lceil \frac{a}{b} \right \rceil -\frac{a}{b}\...
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2 votes
0 answers
31 views

Find the date at which a group of people will reach a target age sum

The goal is to find the date at which the sum of ages of a group of people will reach a certain number. Though I seem to have found an approximative solution, some issues remains, that I don't know ...
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  • 121
1 vote
1 answer
52 views

Hand calculation of divisor summatory function

Maybe this question is stupid, but there was a problem in a math competition (not even in the highest stage) in my county which asked to Find $ \sum_{n\leq390} d(n)$, where $d(n)$ is the number of ...
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5 votes
1 answer
102 views

Prove that the sequence $a, f(a),\cdots$ contains at least one square of an integer

Let $f(n) = n+\lfloor \sqrt{n}\rfloor$. Prove that for every positive integer $a$, the sequence $a, f(a),\cdots$ contains at least one square of an integer. Can someone explain why in the solution ...
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-1 votes
1 answer
21 views

Exponentiated quotient that is also under a floor/ceiling function

$$\biggr \lfloor \frac ab \biggr \rfloor^e $$ Or $$\biggr (\biggr \lfloor \frac ab \biggr \rfloor \biggr )^e$$ I feel like the second option looks worse, but the first option assumes that a floor/...
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1 vote
3 answers
71 views

$\lfloor \underbrace{\sqrt{44...4}}_{2n} \rfloor = \underbrace{66..6}_{n} $

Prove: $\lfloor \sqrt{44} \rfloor = 6 $, $\lfloor \sqrt{4444} \rfloor = 66 $, $\lfloor \sqrt{444444} \rfloor = 666$, ... $$ \lfloor \underbrace{\sqrt{44...4}}_{2n} \rfloor = \underbrace{66..6}_{n} $$ ...
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  • 4,546
0 votes
2 answers
37 views

Floor inequality in proof

I am currently trying to understand the proof that $[-x] = -[x]-1$ if $x$ is not an integer (solutions for $b$.) where $[x]$ is the floor function. Can somebody explain me how he went from $$-n-1 < ...
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2 votes
1 answer
37 views

Find number of points of discontinuity of $f(x)=\frac x5-\left\lfloor \frac x5\right\rfloor+\left\lfloor \frac{x}{2} \right\rfloor$ for $x\in[0,100]$

Find number of points of discontinuity of $$f(x)=\frac{x}{5}-\left\lfloor \frac{x}{5} \right\rfloor+\left\lfloor \frac{x}{2} \right\rfloor$$ for $x\in[0,100]$ . My Attempt The floor function is ...
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  • 7,372
1 vote
2 answers
63 views

Proof $ \lfloor{x}\rfloor + \lfloor{y}\rfloor \le \lfloor{x + y}\rfloor $

I want to prove that $ \lfloor{x}\rfloor + \lfloor{y}\rfloor \le \lfloor{x + y}\rfloor $, I started proving but I got stuck. Let $ x, y \in \mathbb{R} $. Therefore: $ \lfloor{x}\rfloor \le x $ $ \...
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2 votes
0 answers
92 views

A method to find the horizontal asymptote of a composition of the floor function

In an attempt to solve this question, I managed to rewrite the given limit as $$\frac{1}{\sqrt{2}}\lim_{x \to \infty} \left(\pi \lfloor x \rfloor^{3/2} - 2\sqrt{\lfloor x \rfloor} - \frac{4}{\sqrt{\...
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0 votes
2 answers
70 views

Finding the Ceiling function of $(\sqrt{3}+ 1)^{2n}$

Ceiling function of $(\sqrt{3}+ 1)^{2n}$ is $(\sqrt{3} + 1)^{2n} + (\sqrt{3} - 1)^{2n}$. While solving a problem that states $2^{n+1}$ divides ceiling function of $(\sqrt{3}+ 1)^{2n}$. I went through ...
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0 answers
42 views

How to prove that $ \sum_{n=1}^{\infty} \frac{(-1)^{[\log n]}}{n}$ diverges? [duplicate]

How to prove that $\sum_{n=1}^{\infty} \frac{(-1)^{[\log n]}}{n}$ diverges? I've tried something with the following $$\log n - 1 <[\log n] \leq \log n$$ but I haven't got anything from that. Can ...
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  • 353
7 votes
2 answers
361 views

Find $\int_1^\infty \frac{\{x\}}{x(x+1)}dx,$ where $\{x\}$ means $x - \lfloor x \rfloor$.

Question: Find $\int_1^\infty \frac{\{x\}}{x(x+1)}dx,$ where $\{x\}$ means $x - \lfloor x \rfloor$. I have attempted to split this into two integrals, namely $$\int_1^\infty \frac{x}{x(x+1)}dx - \...
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5 votes
1 answer
74 views

Is $(1+c^2)^n-\lfloor(1+c^2)^{n/2}\rfloor^2<(1+c^2)^{(n+1)/2}$ true for all integers $c>1$, when $n$ is an odd integer?

Let $n$ be an odd integer. Is $$(1+c^2)^n-\lfloor(1+c^2)^{n/2}\rfloor^2<(1+c^2)^{(n+1)/2}$$ true for all integers $c>1$? Notes: $c=1$ has a counterexample $2^{31}-\lfloor2^{31/2}\rfloor^2>2^{...
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2 votes
3 answers
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Solve for x: $\lfloor x\rfloor \{\sqrt{x}\}=1$ where $\{x\}=x-\lfloor x\rfloor$

Solve for x: $\lfloor x\rfloor \{\sqrt{x}\}=1$ where $\{x\}=x-\lfloor x\rfloor$ My Attempt: I took intervals of $x$ as $x\in (2,3)$ so $\sqrt x\in (1,2)$. Due to which $\lfloor x\rfloor=2$ and $\{\...
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  • 7,372
0 votes
1 answer
178 views

Recursive analytic continuation of Riemann zeta function

If you read pages 51-55 of the book The Theory Of Functions by Konrad Knopp (Publication date 1947) and you are patient enough to overcome a very bad made digital copy of the book (I do not understand ...
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1 vote
0 answers
29 views

> Estimation of number of distinct integers in the sequence $a_i =\lfloor \frac{n}{n + 1 - i} \rfloor, i\geq 0$ for some given $n$

The problem: Estimation of number of distinct integers in the sequence $a_i =\lfloor \frac{n}{n + 1 - i} \rfloor, i\geq 0$ for some given $n$ This problem is derived from a programming problem that ...
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1 vote
1 answer
115 views

Formulas of Mellin inversion theorem that involve Riemann zeta function $\zeta (s)$ and floor function $\lfloor x\rfloor$

Functions $f(x)=\lfloor x\rfloor$ and $g(s)=\frac{\zeta (s)}{s}$ are related by Mellin inversion theorem, for $c>1$, $\Re(s)>1$. $$\mathcal{M}_x(f(x))(s)=\mathcal{M}_s^{-1}(g(s))(x)$$ $$\tag{1.1}...
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3 votes
2 answers
122 views

Evaluate $\int_0^1 \left(\{ax\}-\frac{1}{2}\right)\left(\{bx\}-\frac{1}{2}\right) dx$ for $a,b\in\mathbb{Z}^+$

Evaluate $$\int_0^1 \left(\{ax\}-\frac{1}{2}\right)\left(\{bx\}-\frac{1}{2}\right) dx$$ where $a$ and $b$ are positive integers and $\{x\}:=x-\lfloor x\rfloor$ WLOG, we may assume that $a\le b$. Via ...
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1 vote
1 answer
41 views

Proving $\frac{n}{r+\delta_1-1}\lfloor\frac{r+\delta_1-2}{r}\rfloor-\frac{n}{r+\delta_2-1}\lfloor\frac{r+\delta_2-2}{r}\rfloor\geq0$, with conditions

Given $n,r,\delta_1,\delta_2$ with $n > r$, $\delta_1 > \delta_2$, both $\delta_1,\delta_2$ are $\geq 2$ and both $r+\delta_1-1, r+\delta_2-1$ divides $n$. I want to prove that \begin{equation*} ...
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1 vote
1 answer
53 views

Show ${\sum_{i=1}^{\frac{p-1}{2}} \lfloor \frac{2iq}{p} \rfloor + {\sum_{i=1}^{\frac{q-1}{2}}} {\lfloor \frac{2ip}{q} \rfloor}}=(p-1)/2\cdot(q-1)/2$

Let $E = {\sum_{i=1}^{\frac{p-1}{2}}} {\lfloor \frac{2iq}{p} \rfloor + {\sum_{i=1}^{\frac{q-1}{2}}} {\lfloor \frac{2ip}{q} \rfloor}}$ and $E'= \frac{p-1}{2} \frac{q-1}{2}$ Im trying to show that they ...
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  • 161
2 votes
3 answers
67 views

How is the inequality $\displaystyle\sum_{r=1}^k\ \Big\lceil\frac{i-1}{2^r}\Big\rceil \leq i-2+k$ acquired?

Edited: The RHS should be $i-2+k$, not $i-2-k$, I made a typo. Probably needed more sleep. While reading some paper about sorting algorithms, I ran through this: $$\displaystyle\sum_{r=1}^k\ \Big\...
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  • 141
0 votes
0 answers
22 views

Approach to solve ceiling identities

Other than trial and error, is there a way to solve the following problem Show that the smallest integer value of $N$ for which $$\left\lceil\frac{N}{4}\right\rceil = 7$$ is 25 I am not able to find a ...
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  • 261
2 votes
1 answer
130 views

Finding $\text{PV}\int_0^\infty\frac{\sec\left(\pi B(xt-\lfloor xt+\frac12 \rfloor\right)-\sec\left(\pi B(x-\lfloor x+\frac12 \rfloor\right)}{x}dx$

The following problem is proposed by a friend $$\text{PV}\int_0^\infty\frac{\sec\left(\pi B(xt-\lfloor xt+\frac12 \rfloor\right)-\sec\left(\pi B(x-\lfloor x+\frac12 \rfloor\right)}{x}\mathrm{d}x,\quad ...
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  • 22.2k
2 votes
0 answers
79 views

prove that $\lfloor nx \rfloor \ge \sum_{i=1}^n \frac{\lfloor xi\rfloor}{i}$.

Prove that for any real number $x$ and for any positive integer $n, \lfloor nx \rfloor \ge \sum_{i=1}^n \frac{\lfloor x i\rfloor}{i}$. I found a solution to this problem, but why is f constant and ...
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  • 1,031
2 votes
1 answer
45 views

Equivalent of floor division in a group of integers mod N.

I'm working on a small programming project, and I'm struggling a bit with calculating fractions of numbers in a commutative group. I'm by no means a mathematician or a programmer, so please bear with ...
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2 votes
0 answers
54 views

How to solve discrete double summation consisting floor function

I am new to math stack exchange and it is my first question. I know how to solve discrete double summation without floor function. Main problem : $$\sum_{x=1}^{\lfloor(n-1)/7\rfloor} \sum_{y=1}^{\...
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0 votes
1 answer
34 views

An inequality involving box function [closed]

Let $t$ be an even integer greater than or equal to $6$ and $n$ is an odd natural number. Then is it always true that : $\lfloor\frac{5-t}{2} \rfloor+1 + \frac {t+n-3}{2} \geq 2$? I have tried for ...
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  • 911
1 vote
1 answer
64 views

How to apply the floor function in algebra?

I was reading a statistics paper and saw a formula with floor operators. I wondered how to solve for one of the variables in the formula, but realized that I did not know how to work with these things....
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  • 189
2 votes
0 answers
70 views

Closed form of $\sum _{k=1}^{n }\:\lfloor{n/k}\rfloor$ [duplicate]

I came across a question in which, if I am able to calculate this sum, $\sum_{k=1}^{n }\:\lfloor{n/k}\rfloor=$ ? it would get solved quite easily. I have never seen any closed form for this question,...
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  • 1,100
0 votes
0 answers
21 views

Best curve fitting - parameters tuning of floored function

I recently came across the mathematical modelling field of curve fitting for a project, so I'm pretty new to this concept. What I'm trying to do is implement an automated way of determining a set of ...
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0 votes
1 answer
54 views

Are the [x] and {x} functions defined for negative numbers? [duplicate]

[x] is the whole part (floor) of x {x} is the fractional (decimal) part of x ...
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0 answers
15 views

How to solve $a + b \lceil cx+d \rceil (e + f \lceil cx + d \rceil) + gx = 0$

I am trying to solve an equation of the following form (derived from a physics problem): $a + b \lceil cx+d \rceil (e + f \lceil cx + d \rceil) + gx = 0$ I can't find a good way to do this, but I'm ...
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0 votes
0 answers
17 views

Question from Polya and Szego's Problems in Analysis concerning the limit of floor function

For the following question: Let $\theta$ be an irrational number, $0< \theta < 1$ and let $g_{n}=0$ or $1$, according as $\lfloor n\theta \rfloor$ and $\lfloor (n-1)\theta \rfloor$ are equal or ...
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