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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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6
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138 views

Explicit definition of a sequence

Suppose there are 6 sequences $a=(a_n)_{n\geq 0}, b=(b_n)_{n\geq 0},c=(c_n)_{n\geq 0},d=(d_n)_{n\geq 0},e=(e_n)_{n\geq 0},f=(f_n)_{n\geq 0}$, the data can be seen here: Data. I found out by trial and ...
5
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0answers
395 views

Show that $f(x) = \lfloor \sqrt{2} \cdot x\rfloor$ is primitive recursive function.

Trying to wrap my head around primitive recursive functions. Especially bounded minimalisation and how to prove something is indeed primitive recursive. Found the following problem for the subject ...
5
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203 views

Binomial Coefficient Identity, Double Series, Floor Function

Show that for any natural numbers $m$ and $n$ such that $ m \le n $ that: $$ \sum_{i=0}^{n}{ \sum_{j=0}^{m}{ \left(-1 \right)^{\lfloor \frac{i}{2} \rfloor+j}2^{n-i}\binom{n-\lfloor \frac{i+1}{2} \...
5
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596 views

Is there a closed form of the sum $\sum _{n=1}^x\lfloor n \sqrt{2}\rfloor$

I need to find the n-th partial sum of: $$\sum _{n=1}^x\lfloor n \sqrt{2}\rfloor$$ Or this sum of a Beatty sequence. I tried to expand as the following: $$=\frac{\sqrt{2} x \left(x+1\right)}{2}-\...
4
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0answers
76 views

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\...
4
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0answers
58 views

Alternating harmonic series containing floor function

Let $$S\left(a\right)=\sum_{n=0}^\infty \frac{\left(-1\right)^{\left[na\right]}}{\left[na\right]+1}$$ where $a\gt 0$, and $\left[\cdot\right]$ denotes the floor function. Consider $a\in \mathbb{Q}$, ...
4
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0answers
128 views

Evaluating $\int_0^{10\pi}(\lfloor \sec^{-1}x\rfloor + \lfloor \cot^{-1}x\rfloor)\,\mathrm{d}x$

My book has this problem: $$\int_0^{10\pi} (\lfloor \sec^{-1}x\rfloor + \lfloor \cot^{-1}x\rfloor) \;dx = \;?$$ Now, the domain of $\sec^{-1}x$ is $({-\infty}, -1) \bigcup (1, \infty)$. How can you ...
4
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1answer
116 views

Disprove $m!=100x^2+20x$ using an estimation for factorial.

$\newcommand{\floor}[1]{\lfloor #1 \rfloor}$ I have the equation $m!=100x^2+20x$ where $x$ and $m$ are real non-negative integers. I wish to disprove for when $m\geq20$ how can I do this? I had an ...
4
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0answers
50 views

Prove or disprove: $\phi: \mathbb{N} \to \mathbb{N}\text{, }\phi(n) = \lfloor n \cdot | \sin( \sqrt{2} \cdot n ) | \rfloor$ is surjective

How do I go about proving or disproving that the following function is surjective? Is there some sort of standard trick for integer functions? $$\phi: \mathbb{N} \to \mathbb{N} \\ \phi(n) = \lfloor ...
4
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76 views

Find $\sum_{k=1}^nk\lfloor k\varphi\rfloor$, where $\varphi$ is golden ratio

I've got this far, which is nothing really. Assuming $S(n)=\sum_{k=1}^n\lfloor k\varphi\rfloor$, for which we have a recursive formula (see here: Solve summation $\sum_{i=1}^n \lfloor e\cdot i \rfloor ...
4
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510 views

Bounds on floor summation $\sum^n_k \lfloor c \cdot k \rfloor $

I've seen a couple posts recently about finding closed forms of or calulating $ \sum^n_k \lfloor c \cdot k \rfloor $ for any real $c>0$. And I was wondering about bounds we can give to this ...
4
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122 views

Solve : $\displaystyle\sum_{k=1}^{n} {\sum_{x=1}^{2^k} \bigg\lfloor\bigg(\dfrac{k}{1 + \pi(x)}\bigg)^{1/k}\bigg\rfloor}$?

Basically, I've been doing some problems involving primes and summation of consecutive primes. I know that there is the imprecise equation for this where $p_1$+$p_2$+$p_3$...+$p_n$ is approximately ...
4
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1answer
100 views

Experiment:-$\sum_{j=1}^{2n+1}(j+1)\left\lfloor {j\cdot j!\over j+1} \right\rfloor$

Experiment of Mathematica to $(1)$ $$\sum_{j=1}^{2n+1}(j+1)\left\lfloor {j\cdot j!\over j+1} \right\rfloor \tag1$$ $\pi(n)$;Prime-counting function $\lfloor x \rfloor$;Floor function We ...
4
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1answer
55 views

Find all real $x$ such that $1990[x] +1989[-x]=1$ (where $[x]$ is the floor function for $x$).

Find all real $x$ such that $1990[x] +1989[-x]=1$ (where $[x]$ is the floor function for $x$). My effort Rearranging our equation we have : \begin{array}{c} 1990[x]+1989[-x]&=1 \\ 1989([x]+[-...
4
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1answer
330 views

Closed form solution for $\left \lfloor{\frac{a}{x}}\right \rfloor$ = $\left \lfloor{\frac{b}{x}}\right \rfloor$?

I need to find the smallest value of $x$ such that: $\left \lfloor{\frac{a}{x}}\right \rfloor$ = $\left \lfloor{\frac{b}{x}}\right \rfloor$ EDIT: where $0 < x < a < b$, and $x \in \mathbb{...
3
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45 views

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}(...
3
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56 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 ...
3
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85 views

Finite double sum $\sum_{k=0}^N\sum_{l=0}^M\left\lfloor\frac{k+l}{c}\right\rfloor$; any advanced summation technique?

Let $M,N,c$ be positive integer. It was astonishing when trying to solve $\sum_{k=0}^N\sum_{l=0}^M\left\lfloor\frac{k+l}{c}\right\rfloor$ to obtain this rather complex looking result \begin{align*} ...
3
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1answer
86 views

How to compute density function of $Y=X-\lfloor X \rfloor$?

We're given that X is a continous r.v. and has a density function of $f_X(x)$ and we're asked to find the density function of $Y=X-\lfloor X \rfloor$ in terms of $f_X(x)$. We're also given the hint : ...
3
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1answer
102 views

Closed-form for Floor Sum 3 - With knowledge of inner expression

Consider the following sum: $$\sum_{k=0}^{n}\left\lfloor\sqrt{k^2+N}\right\rfloor$$ Assume we know the factorization of $N$, in other words, we know for which $k$, $k^2+N$ will be square, according ...
3
votes
1answer
111 views

express series exactly as sum of integrals (geometric insight?)

Suppose for a series there exists a function such that $a_n = f(n)$, then even for a non-monotonically decreasing function: $$A. \sum_{i=n_1+1}^{n_2}f(n)=\int_{n_1}^{n_2}f(x)dx\ +\int_{n_1}^{n_2}(x-\...
3
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104 views

A problem on floor function

Find $m,n\in \mathbb{Z}$ s.t. $$ \sum\limits_{k = 0}^{mn - 1} {\left( { - 1} \right)^{\left\lfloor {\frac{k} {m}} \right\rfloor + \left\lfloor {\frac{k} {n}} \right\rfloor } } = 0 $$ See here for ...
3
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161 views

Correlation between the weak solutions of a differential equation and implied differential equations

Yes,this is very similar to a previous question I asked. That was about normal solutions and not weak solutions. We define the operator known as the implied derivative denoted as $I(f)(x)(g)$ to be: ...
3
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1answer
71 views

Equality using floor function

Let $n\in \mathbb{N}$. How we can show this : $$\lfloor \sqrt{n}+\sqrt{n+1}+\sqrt{n+2}+\sqrt{n+3}\rfloor =\lfloor \sqrt{16n+20}\rfloor$$ by using the concavity of $x\longmapsto \sqrt{x}$. I read an ...
3
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0answers
51 views

Distribution and convergence of the r.vs.: $X_n= \frac{ \lfloor nX \rfloor}{n}$

$X$ is an absolutely continous random variable, with a continous density function, and: $$X_n= \frac{ \lfloor nX \rfloor}{n}$$ What is the distribution of $X_n$, and what can we say about its ...
3
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1answer
225 views

Is there a cycle for modulo operation on a floor?

I have a floor $${\left\lfloor\frac{n}{i}\right\rfloor},$$ where i varies from 1 to n (n can be upto $10^{10}$), and there's another number given 'm' (m upto $10^{5}$). Does (${\left\lfloor\frac{n}{i}...
3
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0answers
430 views

Full series expansion of the floor function

We know if $x$ is not an integer we have $$\left \lfloor x \right \rfloor=x-\frac{1}{2}+\frac{1}{\pi }\sum_{k=1}^{\infty}\frac{\sin(2\pi kx)}{k}$$ Is there an series expansion of floor function ...
3
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0answers
236 views

Finding sum for function with floor-function

I am trying to find a formula to calculate the following sum: $$\sum_{x=0}^n (2ax - {1 \over 2}a^2 - {1 \over 2} a) $$ where $$ a = \left\lfloor {x \over \phi^2} \right\rfloor $$ and $$\phi = {1 + ...
3
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2answers
124 views

Indefinite integral from floor of $x$

I want to calculate $\displaystyle \int [x] dx$ where $[]$ is floor function. I don't know if it is just $kx+C$ for $x \in [k, k+1)$ and $(k+1)x+C$ for $x =k+1$ or I'm missing something
2
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0answers
28 views

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 ...
2
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0answers
58 views

$ \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}^{...
2
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0answers
80 views

Calculate the number of nonnegative integer solutions of $ax+by\leq c$.

If $a$, $b$, and $c$ are known, and $x$ and $y$ are integers greater than or equal to zero, how many possible values of ($x$, $y$) exist that satisfy the equation $$ax + by \le c\,?$$ I have ...
2
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0answers
62 views

Using the 2nd Chebyschev function for describing sums of primes

The question is as follows: Show that for any $n\ge1$, we have $$\psi(n)=\sum_{p\ge n}\left \lfloor\frac{\log n}{\log p}\right \rfloor \log p$$ where $\psi(x)=\sum_{p^m\le x}\log p$, where the sum ...
2
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0answers
83 views

Nested Floor/Ceiling Function for non-Integer Divisors

I am trying to figure out how to find the Floor and Ceiling of a nested Multiplication of fractions. I understand that $ \left\lceil{\frac{x}{mn}}\right \rceil= \left\lceil{\frac{\lceil{\frac{x}{...
2
votes
2answers
97 views

Finding a nonzero polynomial involving the floor function

Find a nonzero polynomial $P(x,y)$ where the coefficients are integers such that $P(\lfloor a \rfloor, \lfloor 2a \rfloor) = 0$ $\forall a \in$ $\mathbb{R}$
2
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0answers
75 views

Limit of trigonometric function involving floor function

I am trying to find the limit of the sequence $$ \frac{\sin(2\pi k\lfloor (n-1)/2\rfloor/n)}{2\sin(\pi k/n)} $$ as $n\to\infty$, where $k\ne0$ is a fixed integer and $\lfloor\cdot\rfloor$ is the ...
2
votes
1answer
68 views

Limit $\lim_{x \rightarrow ( -\frac{1}{10} )^-} [ \frac{1}{x} ]=?$

Find the limit: $$ \lim_{x \rightarrow ( -\frac{1}{10} )^-} [ \frac{1}{x} ]=?$$ $[x]$: floor function my try : $$\frac{1}{x}-1<\lfloor \frac{1}{x}\rfloor\le \frac{1}{x}$$ $$\lim_{x \...
2
votes
1answer
57 views

How can I solve this comparsion between sums?

$Suppose\;m\;intergal\;and\;m\ge2$ $Which \;of\; these \;sums\; is\; asymptotically\; closer \;to\; the\; value\; log_mn!?$ $ \sum_{k=1}^n\lfloor\;log_m k\;\rfloor$ $Or$ $\sum_{k=1}^n\lceil\;log_m ...
2
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0answers
35 views

How to compute$\lfloor k^{\frac{m}{n}}\rfloor$?

Given accurate floating point arithmetic, how to calculate $$a=\lfloor k^{\frac{m}{n}}\rfloor$$ in an efficient way? Here $k,m,n$ are positive integers and $\frac{m}{n}$ and $k^{\frac{m}{n}}$ are ...
2
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0answers
57 views

Definite integral involving floor and fractional part

For a real number $ x$ ,let $[ x]$ denote the largest integer less than or equal to $x$ and $ (x)=x-[x]$.Let $n$ be a positive integer.Then$$ \int_0^ncos(2\pi [x](x))dx$$ is equal to---- I broke down ...
2
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0answers
63 views

On simple calculations with $\sum_{k=1}^{n-1}\left\lfloor{\frac{km}{n}}\right\rfloor=\frac{1}{2}(m-1)(n-1)$ and primes $m$ and $n$

Wikipedia's article for Floor and ceiling functions, relates in the section of Quotients that the identity $$\sum_{k=1}^{n-1} \left\lfloor{\frac{km}{n}}\right\rfloor=\frac{1}{2}(m-1)(n-1)$$ holds ...
2
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0answers
102 views

Find all integers $p>q\ge 0$, such that for all real $x,y \in [0;1]$ following inequality holds $[px]+[py]\ge [qx+y]+[x+qy]$

Find all integers $p>q\ge 0$, such that for all real $x,y \in [0;1]$ following inequality holds $$\lfloor px \rfloor + \lfloor py\rfloor \ge \lfloor qx+y\rfloor+\lfloor x+qy \rfloor$$ I used $x\le ...
2
votes
0answers
79 views

Does $g$ map $\mathbb{R}$ onto the Cantor set?

For $x\in\mathbb{R}$ define \begin{equation} g(x)=1+\tfrac{3}{2}\sum_{k=0}^{\infty}\left(\frac{\left\lfloor2^{2k}x\right\rfloor}{2^{2k}}-\frac{\left\lfloor2^{2k+1}x\right\rfloor}{2^{2k+1}}\right) \...
2
votes
0answers
23 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 ...
2
votes
1answer
326 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 4\left\lfloor\frac{x}{6}\...
2
votes
2answers
78 views

Limit of $f(x)=x-\lfloor x \rfloor$ $\epsilon-\delta$

For $x\in \mathbb{R}$, let $\lfloor x \rfloor$ denote the largest integer that is less than or equal to $x$. For example, $\lfloor 3 \rfloor=3$ and $\lfloor \pi \rfloor=3$. Define $f:\mathbb{R}\to \...
2
votes
0answers
130 views

Integral representations of $\zeta(s)$ using the floor/frac functions. How could this one be derived?

Browsing the web, I found quite a few integral representations for $\zeta(s)$ that use the Fractional part {x} or the Floor-function $\lfloor x\rfloor$ e.g.: $$\zeta(s) = \dfrac{s}{s-1} - \frac12+s \...
2
votes
0answers
73 views

How to pigeonhole the primes between $p_n$ and $p_{n+1}^2$ for twin prime conjecture?

For any full list of the primes up to the $n$th prime: $P = \{2, 3,5,\dots, p_n\}$, any natural number $q$ such that $ p_n \lt q \lt p_{n+1}^2$ that is not sieved by a prime in $P$ is also a prime. ...
2
votes
0answers
2k views

Sum of Floor of Square Root: $S = \sum_{k=1}^{n} \lfloor \sqrt{k}\rfloor$

$$ S = \sum_{k=1}^{n} \lfloor \sqrt{k}\rfloor. $$ Hello, I´m trying to solve this summation. I was able to get $$ a_n = 2a_{n-1} - a_{n-2} $$ for non perfect square numbers and $$ a_n = 2a_{n-1} - a_{...
2
votes
0answers
76 views

Solving an inequality involving sum of floors

Suppose I want to find $t_{critical}(u)$, the least $t\in\mathbb{R}^+$ for a given $u\in\left(0\ldots\dfrac{1}{s}\right]$ satisfying $$f(t)=\lfloor rt\rfloor x+\lfloor s (t-u)\rfloor y + y > h$$ ...