Used for questions and equations involving the floor function, which is defined to be the function that returns the largest integer less than or equal to $x$ (often denoted by $\lfloor x\rfloor$). See also (ceiling-function).

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1answer
23 views

How does this faulty system of integration change the nature of jump discontinuity?

Let's define a sort of faulty integral. For the purposes of this question we shall assume that this is the regular integral. This integral integrates all functions properly however it's gets confused ...
0
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1answer
12 views

Removing Discontinuity in 3-space without changing the partial derivative

Is it possible to find a version of the function $$f(x,y) = x\cdot \lfloor y \rfloor + \lfloor x\rfloor^2$$ That is continuous. ANY operation is allowed in changing the function as long as the ...
1
vote
1answer
16 views

How to bound a ratio of integer and a real number in floor function?

I have : $$n\le\left\lfloor \dfrac{m}{x}\right\rfloor,$$ where $n$, $m$ are positive integers and $x<1$ is a positive real number. I would like to bound the ratio $\frac{n}{m}$. So I will get: ...
2
votes
1answer
18 views

Limit of a floor sum

How can i prove that $ \forall x \in \mathbb{R} \displaystyle \lim_{n \to \infty} \dfrac{\left \lfloor{x}\right \rfloor+\left \lfloor{2x}\right \rfloor+\cdots+\left \lfloor{nx}\right \rfloor}{n^2} = ...
0
votes
2answers
25 views

Relation between a floor and a ceiling function for a problem

I was trying to formulate some problem. I want to find a relation between a floor and ceiling function. Suppose the Property 1 satisfies that it has $\lfloor \frac{n}{2} \rfloor$ number of $X$. Then ...
2
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3answers
120 views

Integral: $\int_{0}^{x}\lfloor\dfrac{1}{1-t}\rfloor dt$

I'm working on an integral problem(the rest of which is irrelevant) and this integral arises, which has stumped me. $$\int_{0}^{1}\int_{0}^{x}\left\lfloor\dfrac{1}{1-t}\right\rfloor dt dx$$ ...
2
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1answer
33 views

Proving an identity involving floor function

Prove that : $$\left \lfloor \dfrac{2 a^2}{b} \right \rfloor - 2 \left \lfloor \dfrac{a^2}{b} \right \rfloor = \left \lfloor \dfrac{2 (a^2 \bmod b)}{b} \right \rfloor $$ Where $a$ ...
2
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2answers
23 views

$ \lim_{x\rightarrow a} f(x)= \lim_{x\rightarrow a} [f(x)]$ then at $x=a$ is there a maxima or minima?

$$ \lim_{x\rightarrow a} f(x)= \lim_{x\rightarrow a} [f(x)]$$ Where [.] denotes the greatest integer function (floor) function. $f(x)$ is non-constant in the neighborhood of 'a' and is continuous ...
2
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1answer
33 views

Limit of Series with differences of Floor function

Problem: Evaluate $$ L=\displaystyle \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \left( \lfloor \frac{2n}{k} \rfloor -2\lfloor \frac{n}{k} \rfloor \right)$$. Please help me with this one. I ...
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2answers
52 views

Possible alternative for finding the Area under the floor function (aka, the integral of floor(x))

So, I had to ask myself the question as to what the area under the floor function could possibly be. I started by graphing the basic $\mbox{floor}(x)$ function (I personally use desmos.com for a nice ...
2
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2answers
162 views

What is $\lfloor i\rfloor$?

So, floor is a function that converts a real number to an integer. It rounds down. This makes sense; however, what about complex numbers? I know that depending on the number, it can be split linearly. ...
1
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2answers
35 views

Inequality with floored squareroots

$(\lfloor \sqrt{n}\rfloor +1)^2\ge n+1$, for all $n\in \mathbb{N}$. I have convinced myself that this is true, but would like to see a formal proof.
1
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1answer
62 views

Area enclosed by the curve $\lfloor |x''| \rfloor +\lfloor |y''| \rfloor = 2$

The area enclosed by the curve $$\bigg\lfloor \frac{|x-1|}{|y-1|}\bigg\rfloor +\bigg\lfloor \frac{|y-1|}{|x-1|}\bigg\rfloor = 2\;,$$ Where $-2 \leq x,y\leq 0$ $\bf{My\; Try::}$ Let $x-1=x'$ and ...
1
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2answers
43 views

When does $\lfloor\sqrt{x^2 + n}\rfloor = \lfloor\sqrt{(x-1)^2 + n}\rfloor$?

Call $x \in[\sqrt{n},n/2] \cap \Bbb{Z} $ a critical point if the following holds $$\lfloor\sqrt{x^2 + n}\rfloor = \lfloor\sqrt{(x-1)^2 + n}\rfloor$$ It appears that mostly this does not happen. For ...
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1answer
40 views

Solve $\lfloor x \rfloor = ax+1$ for integral $a$

Solve $\lfloor x \rfloor = ax+1$, where $a$ is an integer. I have found the values of $x$ for $a=0$, $a=1$ and $a=-1$. But I don't know how to continue. How can I find the solutions for integral ...
1
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2answers
38 views

Summation over a floor function of a first degree polynomial

I've been trying to solve a difficult programming question for the last four days. I've gotten most of it done, but the piece I can't seem to figure out is this: Find a closed form expression of ...
4
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3answers
47 views

Sum of number-of-divisors function equals $\sum_{j=1}^{n} \lfloor n/j \rfloor$.

I am trying to prove the identity $$t(1) + t(2) + \cdots + t(n) = \Big\lfloor \dfrac{n}{1} \Big\rfloor + \Big\lfloor \dfrac{n}{2} \Big\rfloor + \cdots + \Big\lfloor \dfrac{n}{n} \Big\rfloor,$$ where ...
1
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1answer
20 views

Proof of $\sum_{j=1}^{p-1} \lfloor jq/p \rfloor = \frac{1}{2}(q-1)(p-1)$ Involving Pairing of Summands

I've seen the proof of the identity $$\sum_{j=1}^{p-1} \lfloor jq/p \rfloor = \frac{1}{2}(q-1)(p-1)$$ where $p$ and $q$ are coprime positive integers. This involves counting the remainders ...
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3answers
43 views

Moving terms outside of a floor

If I have $x^{\lfloor\frac{c}{a-b}\rfloor}$ is this equivalent to $(x^{\lfloor\frac{1}{a-b}\rfloor})^{\lfloor c \rfloor}$ if a,b,c are all integers and x is between 0 and 1? I'm concerned with this ...
2
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0answers
18 views

How to prove $|n-m|\lt d\implies |f(n)-f(m)|\lt e$ for the following condition?

How to prove $\exists n\in\Bbb{N},\forall e\in \Bbb{R^+},\exists d\in\Bbb{R^+},\forall m\in\Bbb{N}, |n-m|\lt d\implies |f(n)-f(m)|\lt e$ if $f(x)=\lfloor \frac{n}{3}\rfloor$. I don't know how to ...
1
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5answers
58 views

How to prove or disprove $\forall x\in\Bbb{R}, \forall n\in\Bbb{N},n\gt 0\implies \lfloor\frac{\lfloor x\rfloor}{n}\rfloor=\lfloor\frac{x}{n}\rfloor$.

How to prove or disprove $\forall x\in\Bbb{R}, \forall n\in\Bbb{N},n\gt 0\implies \left\lfloor\frac{\lfloor x\rfloor}{n}\right\rfloor=\left\lfloor\frac{x}{n}\right\rfloor$. So we want to prove ...
1
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1answer
77 views

Different ways of evaluating $n!$?

I've recently managed to prove the following result and was hoping to know if it already exists? $ \def\lf{\left\lfloor} \def\rf{\right\rfloor}$ $$ \ln(n!) = \sum_{k=1}^{p_k < n}\left( ...
2
votes
0answers
94 views

Find polynomial congruent to function modulo 6

How to find polynomial with rational coefficients, which is congruent to given function (or prove that it doesn't exist)? Particularly, how to find $P(x)$, so $$P(x)\equiv ...
1
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1answer
54 views

Sums involving floor function

I am looking for a direct formula for this sum $$\sum_{k=0}^n \lfloor{\sqrt{n+k}}\rfloor\lfloor{\sqrt{k}}\rfloor$$ Or a method to efficiently compute the sum for large n
1
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0answers
28 views

Floor Function and Euler-Mascheroni Constant

I need to know how to express the following function, $\lfloor{\sqrt{c^2-707}}\rfloor^2=y$ $(s.t.$ $c $ and $y$ are in $N)$, analytically. I think ...
1
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1answer
40 views

Number of values of $x\in [0,\pi]$ where $f(x)=\lfloor 4\sin x-7\rfloor$ is non derivable is?

Number of values of $x\in [0,\pi]$ where $f(x)=\lfloor 4\sin x-7\rfloor$ is non derivable is? $f(x)=\lfloor 4\sin x-7\rfloor=\lfloor 4\sin x\rfloor-7$ I drew the graph of the $f(x)$ and see that ...
1
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1answer
55 views

Prove $\bigg\lfloor\frac{\lfloor x \rfloor }{m}\bigg\rfloor =\bigg\lfloor\frac{x }{m}\bigg\rfloor $

Prove $\bigg\lfloor\frac{\lfloor x \rfloor }{m}\bigg\rfloor =\bigg\lfloor\frac{x }{m}\bigg\rfloor $ where $x\in \mathbb R , x\geqslant 0$ and $m\in \mathbb N$ What I did: Two cases: $x\in ...
2
votes
1answer
28 views

how to write floor function vectors in polar coordinates

let $$\lfloor{x}\rfloor=y$$ And $$z=x-\lfloor{x}\rfloor$$ Plot the following vector in polar coordinates: $$x\hat{\imath}+(y/z)\hat{\jmath}$$ I know that while transforming from cartesian to polar we ...
1
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1answer
35 views

Proof involving Big O and floor

Trying to prove or disprove this (pretty sure it's correct): Let $\mathcal{F}=\{f\mid f:\mathbb{N}\to\mathbb{R}^+\}$ $$\forall f\in\mathcal{F}: \left\lfloor \sqrt{\lfloor f(n)\rfloor }\right\rfloor ...
0
votes
1answer
29 views

Geometric series and floor function

I came out with a geometric sum with a floor function in the upper limit. It looks somehow like this: $$ \sum_{i=0}^{\lfloor 1/p \rfloor} (1-p)^i,\ \ 0<p<1.$$ I would like to study some ...
1
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1answer
57 views

Is there a simple way of proving that $\lfloor\sqrt{n}\rfloor+\lfloor\sqrt{4n+1}\rfloor = \lfloor\frac{3}{2} \lfloor \sqrt{4n+1} \rfloor\rfloor$?

It appears that $$\left\lfloor\sqrt{n}\right\rfloor+\left\lfloor\sqrt{4n+1}\right\rfloor = \left\lfloor\frac{3}{2} \left\lfloor \sqrt{4n+1} \right\rfloor\right\rfloor$$ for all $n \in \mathbb{N}_{0}$, ...
1
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0answers
53 views

Closed form of sum of rounded number

Let, there is two variable $N$ and $D$. Now, I want to find the closed form of the following sum: $$S_n = \sum_{k=1}^N \lfloor{\frac{k}{D}+\frac{1}{2}}\rfloor$$ Its easy to find the closed form by ...
1
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1answer
186 views

Proof that these two expressions are equivalent

I'm looking for algebraic proof that $$ y=\lim_{N \to \infty}\frac{\prod\limits_{n=0}^{N}x-n}{\prod\limits_{n=0}^{N-\lfloor x\rfloor}{x-\lfloor ...
5
votes
2answers
154 views

Is true that $\sum_{k=1}^n\frac{[kx]}{k}\leq[nx]$, for every $x\in\mathbb{R}$ and for every $n\in\mathbb{Z}^+$?

Is true that $\displaystyle\sum_{k=1}^n\dfrac{[kx]}{k}\leq[nx]$, for every $x\in\mathbb{R}$ and for every $n\in\mathbb{Z}^+$? My work: Let $x\in\mathbb{R}$ be given. For every $k=1,\dots,n$, define ...
7
votes
1answer
87 views

Is there a non brute force way to solve this problem?

A friend of mine asked me to prove or disprove that: $$ \left\lceil\frac{2}{2^{1/n}-1}\right\rceil=\left\lfloor\frac{2n}{\ln 2}\right\rfloor \forall n \in \mathbb{Z^+} $$ First of all, I run a ...
0
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1answer
33 views

Justification of a floor function simplification

Consider the expression: $\left\lfloor\frac{k+1}{2}\right\rfloor$, Where $k \in \mathbb{N}$ (natural numbers) How would I show that $\left\lfloor\frac{k+1}{2}\right\rfloor$ is the same thing as ...
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2answers
40 views

Proving that $\lfloor x + k \rfloor=\lfloor x\rfloor + k$ [duplicate]

How to prove that $\lfloor x + k \rfloor=\lfloor x\rfloor + k$ when $k$ is an integer?
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votes
2answers
36 views

If $a\in R$ and the equation $-3(x-\lfloor x \rfloor)^2+2(x-\lfloor x \rfloor)+a^2=0$ has no integral solution,then all possible values of $a$

If $a\in R$ and the equation $-3(x-\lfloor x \rfloor)^2+2(x-\lfloor x \rfloor)+a^2=0$ has no integral solution,then all possible values of $a$ lie in the interval $(A)(-1,0)\cup(0,1)$ $(B)(1,2)$ ...
0
votes
1answer
48 views

Help with equation that uses floor and ceiling functions

I have this equation and I've been stuck with it for a couple of hours. $\lfloor\log_2x\rfloor + 1 = \lceil\log_2(x+1)\rceil$ I've tried using this ceiling property: $\lceil x\rceil = n ...
0
votes
1answer
39 views

What does this floor expression evaluate to?

Consider the expression $$\left\lfloor\frac{2n-1}{2}\right\rfloor\;:$$ does it make sense that this floor function will evaluate to $n$? Or should it be $n-1$?
1
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1answer
60 views

Proof with induction on a sequence

Let $\{a_k\}_{k=0}^\infty$ be a sequence where $a_0 = 0$ $a_1 = 0$ $a_2 = 2$ $\forall k \geq 3, a_k = a_{\lfloor k/2 \rfloor} + 2$ Show that every element of this sequence is even. I am ...
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1answer
70 views

Using Induction to prove a sequence [closed]

How do i prove that $a_{n+1}$ is even? I have tried many things but can't seem to get anywhere in turns of proving. Define the sequence $(a_n)$ as follows $a_0 = 0$, $a_1 = 0$, $a_2 = 2$, $a_n = ...
0
votes
1answer
31 views

What is meant by $[x]$ in this case?

I'm reading a report on the $\space$Gamma function$\space$ and in the first segment it talks about where the following representation of the Gamma function converges: $$\Gamma(x)= \int^{\infty}_{0} ...
2
votes
1answer
92 views

Function that produces sequence 112123123412345…

I'm trying to find a function/formula for $a_n$ such that it produces the sequence $112123123412345$ and so on. I know that one possible way to do this is to find a function like $n-b_n$ where $b_n$ ...
2
votes
1answer
92 views

Iterating through the integerial points of $f(x)$

The floor function $g(x)=\left\lfloor f(x) \right\rfloor$ jumps at points where $f(x)$ is an integer. What I want is a function that gives the $n$-th jump point from $g(0)$. So, for let's say ...
0
votes
2answers
59 views

How to isolate a variable in a floor function?

I hope someone may be able to help me solve the following equation for $y$? $$x=\frac b{50}\left\lfloor{50ay\over b}\right\rfloor$$ I'm trying to isolate $y$ so I can program an Excel file to solve ...
2
votes
2answers
64 views

If $x$, $\{x\}$, $\lfloor x\rfloor$ are in G.P, find $x$.

If $x$, $\{x\}$, $\lfloor x\rfloor$ are in Geometric Progression, find $x$; $x \neq 0$. Here, $\{x\}=x-\lfloor x\rfloor$ Some properties are pretty evident: $$0\leq \{x\} < 1 \tag{1}$$ ...
0
votes
1answer
96 views

For all real numbers $x$, prove $\lfloor x - 2\rfloor = \lfloor x\rfloor - 2$ [closed]

Prove the following statement: For all real numbers $x$, $\lfloor x - 2\rfloor = \lfloor x\rfloor - 2$ I'd appreciate some help with this. All I know is that the floor function $n$ implies : $n ...
0
votes
2answers
53 views

Function with an asymptote at y=-1 and y=1

I'm looking for a function that has two asymptotes parallel to the x-axis. Preferably it should also only cross the x-axis at (0,0) and be built without using any trigonometric functions. Mind you, if ...
0
votes
1answer
38 views

What floor function identity makes this true?

I know that the graph of these two functions is the same: $$(-1)^{\lfloor x\rfloor} = -2\lfloor x\rfloor + 4\left\lfloor\frac {\lfloor x\rfloor}2\right\rfloor + 1$$ Both of them interchange sign at ...