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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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.: $$\zeta(s) = \dfrac{s}{s-1} - \frac12+s \... 0answers 73 views How to pigeonhole the primes between p_n and p_{n+1}^2 for twin prime conjecture? For any full list of the primes up to the nth prime: P = \{2, 3,5,\dots, p_n\}, any natural number q such that  p_n \lt q \lt p_{n+1}^2 that is not sieved by a prime in P is also a prime. ... 0answers 2k views Sum of Floor of Square Root: S = \sum_{k=1}^{n} \lfloor \sqrt{k}\rfloor$$ 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_{...
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 $$f(t)=\lfloor rt\rfloor x+\lfloor s (t-u)\rfloor y + y > h$$ ...