Pub Date : 2023-07-10DOI: 10.7546/nntdm.2023.29.3.474-485
Y. Aliyev
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.
{"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}
Pub Date : 2023-07-03DOI: 10.7546/nntdm.2023.29.3.445-453
Ouarda Bouakkaz, Abdallah Derbal
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 $$ S(x;k,l)=A(k)x +mathcal{O}_{k}left(xexp left( -frac{theta}{2}sqrt{(2ln x)(lnln x)}right) right) left( 0
{"title":"The mean value of the function frac{d(n)}{d^*(n)} in arithmetic progressions","authors":"Ouarda Bouakkaz, Abdallah Derbal","doi":"10.7546/nntdm.2023.29.3.445-453","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.445-453","url":null,"abstract":"Let $d(n)$ and $d^*(n)$ be, respectively, the number of divisors and the number of unitary divisors of an integer $ngeq 1.$ A divisor $d$ of an integer is to be said unitary if it is prime over $frac{n}{d}.$ In this paper, we study the mean value of the function $D(n)=frac{d(n)}{d^*(n)}$ in the arithmetic progressions $ leftlbrace l+mk mid minmathbb{N}^* text{ and } (l, k)=1 rightrbrace;$ this leads back to the study of the real function $xmapsto S(x;k,l)=underset{nequiv l[k]}{sumlimits_{ n leq x}} D(n).$ We prove that $$ 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}
Pub Date : 2023-07-03DOI: 10.7546/nntdm.2023.29.3.454-461
József Sándor
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.
{"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}
Pub Date : 2023-06-19DOI: 10.7546/nntdm.2023.29.3.426-444
S. Sharma, V. K. Bhat
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.
{"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}
Pub Date : 2023-06-06DOI: 10.7546/nntdm.2023.29.3.421-425
D. Lim
In this note, an attempt has been made to generalize the well-known and useful Riordan’s combinatorial identity via a hypergeometric series approach.
本文试图用超几何级数的方法推广著名而实用的赖尔登组合恒等式。
{"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}
Pub Date : 2023-06-05DOI: 10.7546/nntdm.2023.29.3.407-420
Orhan Dişkaya, H. Menken, P. Catarino
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.
{"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}
Pub Date : 2023-06-01DOI: 10.7546/nntdm.2023.29.2.389-401
Efruz Özlem Mersin
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.
{"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}
Pub Date : 2023-05-27DOI: 10.7546/nntdm.2023.29.2.402-406
Jathan Austin
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.
{"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}
Pub Date : 2023-05-23DOI: 10.7546/nntdm.2023.29.2.378-388
J. Sándor, K. Atanassov
Some new results for the maximum and minimum exponents in factorizing integers are obtained. Related functions and generalized arithmetical functions are also introduced.
得到了整数分解中最大指数和最小指数的一些新结果。介绍了相关函数和广义算术函数。
{"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}
Pub Date : 2023-05-18DOI: 10.7546/nntdm.2023.29.2.372-377
K. Ayadi, Chiheb Ben Bechir
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.
{"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}
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