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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40 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
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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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42 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=\...
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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-\...
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1answer
58 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; $d(142)=7$, $d(123785)=8$ ...
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1answer
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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 {...
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1answer
37 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 ...
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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}}\...
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4answers
64 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 ...
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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 ...
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1answer
42 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 $25,2019$?
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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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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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1answer
112 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, ...
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2answers
27 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\...
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2answers
41 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 ...
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4answers
57 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? $g(0.5) = ?$ $g(-0.5) = ?$
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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 ...
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1answer
58 views

Evaluate $\lim_{x \to 0} \left\lfloor(1-e^x)\frac {\sin x}{|x|}\right\rfloor$

How to simplify the following limit:$$ \lim_{x \to 0} \left\lfloor(1-e^x)\frac {\sin x}{|x|}\right\rfloor,$$ where $\lfloor\cdot\rfloor$ represents the greatest integer function. Given limit is in 0/...
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Prove that $(nx-\lfloor nx \rfloor)_{n\geq 1}$ has a finite number of limit points

Prove that $(nx-\lfloor nx \rfloor)_{n\geq 1}$ has a finite number of limit points with $x\in \mathbb{Q}$ and $x>0$. Furthermore $\lfloor .\rfloor:\mathbb{R} \to \mathbb{Z}$ the floor function I ...
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2answers
43 views

Logarithm of factorial equal to sum of logarithm of primes

Let $N$ a positive integer. Denote $\mathcal{P}$ the set of prime numbers. I have to show that \begin{align} \log(N!) = \sum_{p^{\nu}\leq N \\ p\in \mathcal{P}} \left\lfloor\dfrac{N}{p^{\nu}}\right\...
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$ \lfloor (2^{2})/3 \rfloor +… + \lfloor (2^{1000})/3 \rfloor = \frac{2^{A}-B}{C}$, minimum of $A+B+C$?

$$ \lfloor (2^{2})/3 \rfloor + \lfloor (2^{3})/3 \rfloor + \lfloor (2^{4})/3 \rfloor + ... + \lfloor (2^{999})/3 \rfloor + \lfloor (2^{1000})/3 \rfloor = \frac{2^{A}-B}{C},$$ $$ A,B,C \in \mathbb{Z}^{...
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22 views

Upper bound of floor function

I'm studying a paper where the authors stated preliminarily: $$\mid R_{i,T}\mid \leq \frac{C \theta_0}{K(i_\theta,\theta,T)}, \forall i\in\{1,\dotsc,\theta^x\}$$ where $C$ is a finite contant; $\...
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42 views

Find the range of $f(x)=(-1)^{\lfloor x \rfloor} (x-{\lfloor x \rfloor})$

Find the range of $f(x)=(-1)^{\lfloor x \rfloor} (x-{\lfloor x \rfloor})$ I don't know how to work with this function, and how to find the range. I tried to find the domain of its inverse, but i can'...
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1answer
21 views

How to linearize a floor funtion in constraints?

I want to linearize the following constraints: $Y_ij^tb$ ≤ $M$.$X_ij^tb$ $Y_ij^tb$ ≤ ([$S^tb$]+1) $Y_ij^tb$ ≥ ([$S^tb$]+1)-(1-$X_ij^tb$).$M$ $Y_ij$^tb ≥ 0 where a floor function is used. $M$ is ...
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1answer
31 views

Solving an inequality with the floor operator

Assume that $A$, $B$, $C$ are positive real numbers and that $I$ is a positive integer. How could I isolate $A$ in the inequality $\left \lfloor{AB/C}\right \rfloor \geq I$ ? The best I could do: \...
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2answers
63 views

Floor function summation[difficult]

The question is to find the value of — $$\sum_{r=1}^{502} \Big \lfloor \frac{305r}{503}\Big \rfloor$$ The answer is pretty big, so I don't think trial and error will work here. I seriously can't ...
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1answer
30 views

Floor function equation with $n$ solutions

The question is — The equation $\lfloor ax \rfloor = x$ has exactly $n$ distinct solutions, given that $n \in \mathbb{N}, n \geqslant 2$ and $a \in \mathbb{R}, a > 1$. Find the range of $a$. My ...
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1answer
124 views

Evaluating $\sum_{n=0}^{\infty}ne^{1-n}$ using calculus

I'm trying to evaluate the following integral which popped up in MIT Integration Bee 2015 which involves the floor function. $$\int_{0}^{\infty}\left(xe^{1-x}-\lfloor x\rfloor e^{1-\lfloor x\rfloor}\...
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Voyage into the golden screen

We start from A004718 named "The Danish composer Per Nørgård's "infinity sequence", invented in an attempt to unify in a perfect way repetition and variation" $$a(2n) = -a(n), a(2n+1) = a(n) + 1, a(0)=...
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Solve $x + \lfloor y \rfloor + \{ z \} = 13.2$; $\{ x \} + y + \lfloor z \rfloor = 15.1$; $\lfloor x \rfloor + \{ y \} + z = 14.3$

The equation system is — $x + \lfloor y \rfloor + \{ z \} = 13.2$ $\{ x \} + y + \lfloor z \rfloor = 15.1$ $\lfloor x \rfloor + \{ y \} + z = 14.3$ Now I've tried substituting $n$ with $\lfloor n \...
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Maximum and minimum values of $\left\lfloor \frac{x}{nm}\right\rfloor - \left\lfloor \frac{1}{m}\left\lfloor \frac{x}{n}\right\rfloor \right\rfloor$

I feel like I need some additional pointers on the following questions as I am unable to come up with a solution for it: If $m$ and $n$ are any integers, and $x$ is any positive real number, what are ...
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Maximum and minimum value of ⌊2x⌋ − 2 ⌊x⌋ for any real number x?

Working through some sample problems on flooring from a guide book and I'm stuck on the following questions: What is the maximum and minimum value of $\lfloor 2x\rfloor − 2 \lfloor x\rfloor$ for any ...
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1answer
14 views

How to make a transformation to the floor function to the right or left?

Assume a function called $f(x)$ , Then all of us know that of we draw $f(x+a)$ it will be a transformation to the left or right and $f(x)+b$ to up or down. But when I drew floor function on Desmos ...
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1answer
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Prove with geometry: $\lfloor\frac{p}{q}\rfloor+\lfloor\frac{2p}{q}\rfloor+\cdots+\lfloor\frac{(q-1)p}{q}\rfloor=(p-1)(q-1)/2$, for coprime $p$, $q$

The question is to prove this using geometry: For $p$ and $q$ coprime, $$\left\lfloor\frac{p}{q}\right\rfloor +\left\lfloor\frac{2p}{q}\right\rfloor+\cdots + \left\lfloor\frac{(q-1)p}{q}\right\...
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1answer
44 views

Minimum value of a floor function [closed]

Minimum value of a floor function. $\left \lfloor{(a+b+c)/d}\right \rfloor + \left \lfloor{(a+b+d)/c}\right \rfloor +\left \lfloor{(a+d+c)/b}\right \rfloor + \left \lfloor{(d+b+c)/a}\right \rfloor$ My ...
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1answer
43 views

Solve $\int_{2}^{341} \left(x - \lfloor x \rfloor \right)^2$

The question is $\int_{2}^{341} \left(x - \lfloor x \rfloor \right)^2$. I understand how to solve integrals of floor functions (they get converted to discrete integrals) and even just this part: $(x ...
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1answer
96 views

A bizarre expression for cardinality involving summation of roots and floor function

Show that number of triples $(a,b,c)$ with $a,b,c\in [1,n]$ such that $ab=c$ is given by $$\bigl|\bigl\{(a,b,c)\in [1,n]^3:ab=c\bigr\}\bigr|=2\sum_{i=1}^{\left\lfloor\sqrt{n}\right\rfloor}\Big(\...
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46 views

Proof that $\lfloor \log(n) \rfloor = \lfloor \log((n−1)/2) \rfloor + 1$

I have an assignment tomorrow and we should proof that $$\lfloor{\log n}\rfloor = \left\lfloor \log\left( \Big\lfloor \frac{n-1}2 \Big\rfloor \right) \right\rfloor + 1$$ But the floor drives me ...
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1answer
56 views

Can I split this inequality like this?

Recently I had solved this number theory problem but after I solved it I was a bit uncertain whether my approach was correct so I approached AOPS. The problem is : Prove that $[x] + [y] + [x + y] \...
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50 views

Reasoning about remainders and the Möbius function

This one seems counter intuitive to me but I am not seeing a mistake in my reasoning. Please let me know if you find one. Let: $x > 0$ be an integer $\mu(x)$ be the möbius function $x\#$ be the ...
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1answer
71 views

Prove that $ \left \lfloor {\log n} \right \rfloor = \left \lfloor {\log \left \lceil \frac {n-1}{2}\right \rceil} \right \rfloor + 1$

We have been doing algorithm analysis in university, and after analyzing binary search algorithm, the following equation resulted. What we have to do now is to prove that $ \left \lfloor {\log n} \...
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2answers
94 views

Limit of $\left\lfloor x \left\lfloor \frac1x \right\rfloor \right\rfloor$, as $x$ goes to zero

Find $\lim\limits_{x\to 0^+}\left\lfloor x \left\lfloor \frac1x \right\rfloor \right\rfloor$ and $\lim\limits_{x\to 0^-}\left\lfloor x \left\lfloor \frac1x \right\rfloor \right\rfloor$ ? See the ...
2
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1answer
61 views

Counting integers with a least prime factor greater than $x$ in a sequence of $x$ consecutive integers.

It is well known from Sylvester-Schur that in any sequence of $x$ consecutive integers, there is always at least one integer divisible by a prime greater than $x$. I am interested in counting the ...
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1answer
24 views

proving floor, and rounding floats in some programming languages

Here is a simple property which is very useful while programming : In most programming languages, ...
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1answer
25 views

Let $x \in \Bbb R$, $n \in \Bbb N$, show that $n \lfloor x \rfloor \leq \lfloor nx \rfloor \leq n \lfloor x \rfloor + (n-1)$

Problem: Let $x \in \Bbb R$, $n \in \Bbb N$, show that $n \lfloor x \rfloor \leq \lfloor nx \rfloor \leq n \lfloor x \rfloor + (n-1)$ I have one part of the inequality, namely that since $ \lfloor x ...
2
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2answers
61 views

Finding the floor of $\sqrt{d^2-1}$

If we are given $d$ an integer, then I've seen it written that the floor of $\sqrt{d^2-1}=d-1$, but how is this found? It's not immediately obvious, at least not to me.It makes me feel there must be ...
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1answer
80 views

Show that f is continuous at all c ∈ R \ Z and discontinuous at all c ∈ Z.

For any x ∈ R define the floor of x, denoted [x],to be the largest integer y with y ≤ x. Then define a function f : R → R by f(x) =[x]. Show that f is continuous at all c ∈ R \ Z and discontinuous at ...
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2answers
67 views

Reasoning about $\left(\left\lfloor\frac{2x}{i}\right\rfloor -2\left\lfloor\frac{x}{i}\right\rfloor\right)$

I am working on an alternative argument for Bertrand's Postulate that depends on the following argument. Please let me know if I made a mistake or if any point is unclear. Let: $p_k$ be the $k$th ...