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Notes on Number Theory and Discrete Mathematics最新文献

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Digits of powers of 2 in ternary numeral system 三进制数制中2的幂位数
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-07-10 DOI: 10.7546/nntdm.2023.29.3.474-485
Y. Aliyev
{"title":"Digits of powers of 2 in ternary numeral system","authors":"Y. Aliyev","doi":"10.7546/nntdm.2023.29.3.474-485","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.474-485","url":null,"abstract":"We study the digits of the powers of 2 in the ternary number system. We propose an algorithm for doubling numbers in ternary numeral system. Using this algorithm, we explain the appearance of “stairs” formed by 0s and 2s when the numbers $2^n (n=0,1,2, ldots)$ are written vertically so that for example the last digits are forming one column, the second last digits are forming another column, and so forth. We use the patterns formed by the leftmost digits, and the patterns formed by the rightmost digits to prove that the sizes of these blocks of 0s and 2s are unbounded. We also study how this regularity changes when the digits are taken between the left end and the right end of the numbers.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47967839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The mean value of the function frac{d(n)}{d^*(n)} in arithmetic progressions 函数frac｛d（n）｝｛d^*（n）}在算术级数中的平均值
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-07-03 DOI: 10.7546/nntdm.2023.29.3.445-453
Ouarda Bouakkaz, Abdallah Derbal
{"title":"The mean value of the function frac{d(n)}{d^*(n)} in arithmetic progressions","authors":"Ouarda Bouakkaz, Abdallah Derbal","doi":"10.7546/nntdm.2023.29.3.445-453","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.445-453","url":null,"abstract":"Let $d(n)$ and $d^*(n)$ be, respectively, the number of divisors and the number of unitary divisors of an integer $ngeq 1.$ A divisor $d$ of an integer is to be said unitary if it is prime over $frac{n}{d}.$ In this paper, we study the mean value of the function $D(n)=frac{d(n)}{d^*(n)}$ in the arithmetic progressions $ leftlbrace l+mk mid minmathbb{N}^* text{ and } (l, k)=1 rightrbrace;$ this leads back to the study of the real function $xmapsto S(x;k,l)=underset{nequiv l[k]}{sumlimits_{ n leq x}} D(n).$ We prove that $$ S(x;k,l)=A(k)x +mathcal{O}_{k}left(xexp left( -frac{theta}{2}sqrt{(2ln x)(lnln x)}right) right) left( 0<theta<1 right),$$ where $quad A(k)=dfrac{c}{k}prodlimits_{pmid k}left(1+dfrac{1}{2}sumlimits_{n=2}^{+infty}dfrac{1}{p^{n}}right)^{-1}left( c=zeta(2)prodlimits_{p} left(1-dfrac{1}{2p^2}+dfrac{1}{2p^3} right) right).$","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48361990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On certain inequalities for the prime counting function – Part III 关于质数计数函数的某些不等式 - 第三部分
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-07-03 DOI: 10.7546/nntdm.2023.29.3.454-461
József Sándor
{"title":"On certain inequalities for the prime counting function – Part III","authors":"József Sándor","doi":"10.7546/nntdm.2023.29.3.454-461","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.454-461","url":null,"abstract":"As a continuation of [10] and [11], we offer some new inequalities for the prime counting function $pi (x).$ Particularly, a multiplicative analogue of the Hardy–Littlewood conjecture is provided. Improvements of the converse of Landau's inequality are given. Some results on $pi (p_n^2)$ are offered, $p_n$ denoting the $n$-th prime number. Results on $pi (pi (x))$ are also considered.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"44 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139363977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On vertex resolvability of a circular ladder of nonagons 关于非多边形圆形梯形的顶点可分解性
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-06-19 DOI: 10.7546/nntdm.2023.29.3.426-444
S. Sharma, V. K. Bhat
{"title":"On vertex resolvability of a circular ladder of nonagons","authors":"S. Sharma, V. K. Bhat","doi":"10.7546/nntdm.2023.29.3.426-444","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.426-444","url":null,"abstract":"Let $H=H(V,E)$ be a non-trivial simple connected graph with edge and vertex set $E(H)$ and $V(H)$, respectively. A subset $mathbb{D}subset V(H)$ with distinct vertices is said to be a vertex resolving set in $H$ if for each pair of distinct vertices $p$ and $q$ in $H$ we have $d(p,u)neq d(q,u)$ for some vertex $uin H$. A resolving set $H$ with minimum possible vertices is said to be a metric basis for $H$. The cardinality of metric basis is called the metric dimension of $H$, denoted by $dim_{v}(H)$. In this paper, we prove that the metric dimension is constant and equal to $3$ for certain closely related families of convex polytopes.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49329029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A note on a generalization of Riordan’s combinatorial identity via a hypergeometric series approach 用超几何级数方法推广Riordan组合恒等式的一个注记
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-06-06 DOI: 10.7546/nntdm.2023.29.3.421-425
D. Lim
{"title":"A note on a generalization of Riordan’s combinatorial identity via a hypergeometric series approach","authors":"D. Lim","doi":"10.7546/nntdm.2023.29.3.421-425","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.421-425","url":null,"abstract":"In this note, an attempt has been made to generalize the well-known and useful Riordan’s combinatorial identity via a hypergeometric series approach.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45240767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the bivariate Padovan polynomials matrix 关于二元Padovan多项式矩阵
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-06-05 DOI: 10.7546/nntdm.2023.29.3.407-420
Orhan Dişkaya, H. Menken, P. Catarino
{"title":"On the bivariate Padovan polynomials matrix","authors":"Orhan Dişkaya, H. Menken, P. Catarino","doi":"10.7546/nntdm.2023.29.3.407-420","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.407-420","url":null,"abstract":"In this paper, we intruduce the bivariate Padovan sequence we examine its various identities. We define the bivariate Padovan polynomials matrix. Then, we find the Binet formula, generating function and exponential generating function of the bivariate Padovan polynomials matrix. Also, we obtain a sum formula and its series representation.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45583232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Hyperbolic Horadam hybrid functions 双曲霍达姆混合函数
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-06-01 DOI: 10.7546/nntdm.2023.29.2.389-401
Efruz Özlem Mersin
{"title":"Hyperbolic Horadam hybrid functions","authors":"Efruz Özlem Mersin","doi":"10.7546/nntdm.2023.29.2.389-401","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.389-401","url":null,"abstract":"The aim of this paper is to introduce the hybrid form of the hyperbolic Horadam function and to investigate some of its properties such as the generating function. Another aim is to define hyperbolic Horadam hybrid sine and cosine functions and their symmetrical forms. For newly defined functions, some properties such as the recursive relations, derivatives, Cassini and De Moivre type identities are examined. In addition, the quasi-sine Horadam hybrid function and three-dimensional Horadam hybrid spiral are defined.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41530764","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A note on generating primitive Pythagorean triples using matrices 关于使用矩阵生成原始毕达哥拉斯三元组的注释
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-05-27 DOI: 10.7546/nntdm.2023.29.2.402-406
Jathan Austin
{"title":"A note on generating primitive Pythagorean triples using matrices","authors":"Jathan Austin","doi":"10.7546/nntdm.2023.29.2.402-406","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.402-406","url":null,"abstract":"We present matrices that generate families of primitive Pythagorean triples that arise from generalized Fibonacci sequences. We then present similar results for generalized Lucas sequences and primitive Pythagorean triples.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43726910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On certain arithmetical functions of exponents in the factorization of integers 整数分解中指数的若干算术函数
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-05-23 DOI: 10.7546/nntdm.2023.29.2.378-388
J. Sándor, K. Atanassov
{"title":"On certain arithmetical functions of exponents in the factorization of integers","authors":"J. Sándor, K. Atanassov","doi":"10.7546/nntdm.2023.29.2.378-388","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.378-388","url":null,"abstract":"Some new results for the maximum and minimum exponents in factorizing integers are obtained. Related functions and generalized arithmetical functions are also introduced.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44702393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A note on the length of some finite continued fractions 关于有限连分式长度的一个注记
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-05-18 DOI: 10.7546/nntdm.2023.29.2.372-377
K. Ayadi, Chiheb Ben Bechir
{"title":"A note on the length of some finite continued fractions","authors":"K. Ayadi, Chiheb Ben Bechir","doi":"10.7546/nntdm.2023.29.2.372-377","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.372-377","url":null,"abstract":"In this paper, based on a 2008 result of Lasjaunias, we compute the lengths of simple continued fractions for some rational numbers whose numerators and denominators are explicitly given.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42214605","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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