# 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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• 45
1 vote
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 ...
• 65
1 vote
38 views

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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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• 535
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• 1,339
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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 ...
• 75
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### 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 $$\frac{n + \sqrt{n^2+4}}{2},$$ ...
• 667
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### 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 ...
• 121
1 vote
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 ...
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 ...
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/...
• 896
1 vote
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### $\lfloor \underbrace{\sqrt{44...4}}_{2n} \rfloor = \underbrace{66..6}_{n}$

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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### 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 ...
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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• 926
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### 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 ...
• 45
1 vote
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### 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*} ...
1 vote
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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• 22.2k
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### 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 ...
• 1,031
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### 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 ...
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}^{\...
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 ...
• 911
1 vote
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### 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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### 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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### 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 ...
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 ...
### 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 ...
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 ...