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pro vyhledávání: '"11A63"'
Autor:
Sobolewski, Bartosz
Let $\mathsf{s}(n)$ denote the sum of binary digits of an integer $n \geq 0$. In the recent years there has been interest in the behavior of the differences $\mathsf{s}(n+t)-\mathsf{s}(n)$, where $t \geq 0$ is an integer. In particular, Spiegelhofer
Externí odkaz:
http://arxiv.org/abs/2412.15851
The third-named author recently proved [Israel J. of Math. 258 (2023), 475--502] that there are infinitely many \textit{collisions} of the base-2 and base-3 sum-of-digits functions. In other words, the equation \[ s_2(n)=s_3(n) \] admits infinitely m
Externí odkaz:
http://arxiv.org/abs/2412.09124
Autor:
Kim, Jiseong
In this paper, we study the sum of the divisor function over sets with digit restrictions.
Comment: 8 pages, the results are improved
Comment: 8 pages, the results are improved
Externí odkaz:
http://arxiv.org/abs/2411.09076
Autor:
MacVicar, Neil
Let $C$ be the attractor of the IFS $\{f_{d}(z) = (-n+i)^{-1}(z+d): d\in D\}$, $D\subset\{0, 1, \ldots, n^{2}\}$ and let $\dim$ denote the box-counting dimension. It is known that for all $\lambda\in[0, 1]$, that the set of complex numbers $\alpha$ f
Externí odkaz:
http://arxiv.org/abs/2410.19237
In this article, we present a method to construct $e$-power $b$-happy numbers of any height. Using this method, we construct a tree that encodes these happy numbers, their heights, and their ancestry--relation to other happy numbers. For fixed power
Externí odkaz:
http://arxiv.org/abs/2410.13990
We study two positional numeration systems which are known for allowing very efficient addition and multiplication of complex numbers. The first one uses the base $\beta = \imath - 1$ and the digit set $\mathcal{D} = \{ 0, \pm 1, \pm \imath \}$. In t
Externí odkaz:
http://arxiv.org/abs/2410.02418
Autor:
Fox, N. Bradley, Fox, Nathan H., Grundman, Helen G., Lynn, Rachel, Namoijam, Changningphaabi, Vanderschoot, Mary
For a base $b \geq 2$, the $b$-elated function, $E_{2,b}$, maps a positive integer written in base $b$ to the product of its leading digit and the sum of the squares of its digits. A $b$-elated number is a positive integer that maps to $1$ under iter
Externí odkaz:
http://arxiv.org/abs/2409.09863
Autor:
Mittal, Mohit, Sharma, Divyum
We provide an effective upper bound for positive integers with bounded Hamming weights with respect to both a linear recurrence numeration system and an Ostrowski-$\alpha$ numeration system, where $\alpha$ is a quadratic irrational. We prove a simila
Externí odkaz:
http://arxiv.org/abs/2409.06232
Autor:
Minabutdinov, Aleksei
Let $s(n)$ denote the number of "$1$"s in the dyadic representation of a positive integer $n$ and sequence $S(n) = s(1)+s(2)+\dots+s(n-1)$. The Trollope-Delange formula is a classic result that represents the sequence $S$ in terms of the Takagi funct
Externí odkaz:
http://arxiv.org/abs/2407.15201
Autor:
Yakkou, Hamid Ben, Boudine, Brahim
Let $m$ be a rational integer $(m \neq 0, \pm 1)$ and consider a pure number field $K = \mathbb{Q} (\sqrt[n]{m}) $ with $n \ge 3$. Most papers discussing the monogenity of pure number fields focus only on the case where $m$ is square-free. In this pa
Externí odkaz:
http://arxiv.org/abs/2407.00819