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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17
votes
7answers
3k views

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.
-1
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2answers
60 views

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.
0
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2answers
47 views

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\...
0
votes
0answers
36 views

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\...
1
vote
3answers
85 views

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 \...
4
votes
4answers
51 views

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} = \...
0
votes
1answer
23 views

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\...
0
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2answers
31 views

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 ...
3
votes
1answer
115 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{...
1
vote
2answers
84 views

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 ...
-1
votes
1answer
58 views

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 \...
1
vote
0answers
70 views

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{...
3
votes
2answers
104 views

Any suggestions on a closed form of $\sum_{a=1}^{N} \left({\lfloor{\frac{N+1}{a}}\rfloor+\lfloor{\frac{N-1}{a}}\rfloor}\right)$

Now the sum is approximated by $$\sum_{a=1}^{N} \left({\left\lfloor{\frac{N+1}{a}}\right\rfloor + \left\lfloor{\frac{N-1}{a}}\right\rfloor}\right) \approx 2 \sum_{a=1}^{N} \left\lfloor{\frac{N}{a}}\...
9
votes
1answer
187 views

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: ...
3
votes
2answers
205 views

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

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

$\mathcal{I}=\int\limits_0^0 \{x\}^{\lfloor x\rfloor}\,\mathrm dx=0\textrm{ or undefined ?}$

Consider the following integral: $$\mathcal{I}=\int\limits_0^0 \{x\}^{\lfloor x\rfloor}\,\mathrm dx$$ Now, my concern is that at $x=0$, the value of the integrand is $0^0$ which is undefined. It's ...
2
votes
3answers
106 views

Evaluate $ \lim_{x \to -0.5^{-}} \left\lfloor\frac{1}{x} \left\lfloor \frac{-1}{x} \right\rfloor\right\rfloor$

Evaluate $$L=\lim_{x \to -0.5^{-}} \left\lfloor\frac{1}{x} \left\lfloor \frac{-1}{x} \right\rfloor\right\rfloor $$ My try: Let $t=\frac{1}{x}$ Now when $ t \to -0.5^{-}$ we have $t \to -2^{+}$ we ...
10
votes
4answers
1k views

$\lfloor (2+\sqrt{3})^n \rfloor $ is odd

Let $n$ be a nonnegative integer. Show that $\lfloor (2+\sqrt{3})^n \rfloor $ is odd and that $2^{n+1}$ divides $\lfloor (1+\sqrt{3})^{2n} \rfloor+1 $. My attempt: $$ u_{n}=(2+\sqrt{3})^n+(2-\sqrt{3}...
3
votes
3answers
127 views

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? ...
2
votes
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 $$\phi(n) := \lfloor H_n \rfloor $$...
1
vote
3answers
48 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 $$n-1$$.
0
votes
2answers
34 views

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 \...
0
votes
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 ...
0
votes
1answer
57 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 ...
7
votes
1answer
221 views

Compute $\lim_{x\to 0}\frac{x}{[x]}$

When I take left hand limit of the function $\lim\limits_{x\to 0}\frac{x}{[x]}$, then $\lim\limits_{h\to 0^{-}}\frac{-h}{[-h]}=\lim_{h\to 0^{-}}\frac{-h}{-1}=0$ where $0<h<1$ and $[\cdot ]$ is ...
0
votes
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 &...
4
votes
3answers
79 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 ...
1
vote
3answers
55 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 ...
3
votes
2answers
78 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-\...
2
votes
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 ...
0
votes
1answer
43 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
votes
1answer
79 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; $d(142)=7$, $d(123785)=8$ ...
1
vote
1answer
53 views

Concrete mathematics $\sum_{0 \le k \lt n} (\sqrt {\lfloor k \rfloor})$

I've stumbled upon this sum while I was studying Knuth's Concrete Mathematics: $$\sum_{0 \le k \lt n} (\sqrt {\lfloor k \rfloor})$$ The derivation the book makes is $$\sum_{0 \le k \lt n} (\sqrt {...
0
votes
1answer
38 views

Counting how many integers give a integer in the square root [closed]

My question is about a Mathematica code that gives the same as a mathematical experssion (for integers) but I do not know why. The mathematica code is the following (you can try it out for integers ...
1
vote
2answers
43 views

Solve and asymptotic expansion of $\sum_{a=1}^{H} \sum_{b=a+1}^{H} \left\lfloor{\frac{H}{a\, b}}\right\rfloor$

I am solving constrained polynomial systems resulting in constrained sums. I am looking to see if $$\sum_{a=1}^{H} \sum_{b=a+1}^{H} \left\lfloor{\frac{H}{a\, b}}\right\rfloor$$ is expressible in ...
4
votes
4answers
69 views

Prove that $\lfloor N/ \lfloor \sqrt{N} \rfloor^2\rfloor = 1$ for $N > 8$

Prove that $\lfloor N/ \lfloor \sqrt{N} \rfloor^2\rfloor = 1$ for $N > 8$. This is simple to check by listing and the only exceptions are for $N = 2,3,8$. I am looking for a simple proof if ...
1
vote
0answers
28 views

How 'MOD' the binary operation works with floor function?

I was looking for some example on MOD and floor function from Concrete Mathematics book(1). But, the solution of each example is very confusing for me. The general formula of x mod y is; x mod y = x ...
1
vote
1answer
45 views

How to find the week day of (any) given date? [duplicate]

How to find the week day of any given date? Say we need to know in which week-day was June $25,2019$?
3
votes
1answer
58 views

Find $\lfloor z \rfloor $ given that $z=(\{\sqrt{3}\}^2-2\{\sqrt{2}\}^2)/(\{\sqrt{3}\}-2\{\sqrt{2}\})$

Let $\lfloor x \rfloor$ denote the greatest integer function, and $\{x\}=x-[x]$ the fractional part of $x$. If $$z=\cfrac{\{\sqrt{3}\}^2-2\{\sqrt{2}\}^2}{\{\sqrt{3}\}-2\{\sqrt{2}\}}$$ find $\lfloor z \...
0
votes
0answers
38 views

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 ...
3
votes
1answer
117 views

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, ...
0
votes
2answers
51 views

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 $f(n)=\frac{x}{n}$ ...
1
vote
2answers
1k views

Ceil () and Floor()

I already know the basic rules for the both functions: $$\text{Ceil}(2.5)=3\\ \text{Floor}(2.5)=2$$ But I could not understand the following these: $$\text{Ceil}(2.6, 0.25)=2.75\\ \text{Floor}(2.6, ...
0
votes
2answers
28 views

Solving equations containing floor functions with rational terms

So, every question in here regarding these kinds of equations only involves natural numbers. Generally, I'm looking for a method to solve equations on the form: $\lfloor ax+b\rfloor=c$ for $a,b\in\...
15
votes
2answers
534 views

Efficient computation of $\sum_{k=1}^n \lfloor \frac{n}{k}\rfloor$

I realize that there is probably not a closed form, but is there an efficient way to calculate the following expression? $$\sum_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor$$ I've noticed $$\sum_{k=...
3
votes
2answers
184 views

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 ...
-1
votes
4answers
63 views

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? $g(0.5) = ?$ $g(-0.5) = ?$
4
votes
3answers
122 views

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^{...
0
votes
1answer
50 views

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 ...