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On tertions and other algebraic objects 关于叔位和其他代数对象
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-12-29 DOI: 10.7546/nntdm.2023.29.4.861-880
K. Atanassov
{"title":"On tertions and other algebraic objects","authors":"K. Atanassov","doi":"10.7546/nntdm.2023.29.4.861-880","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.861-880","url":null,"abstract":"The concept of the object called “tertion” is discussed. Some operations over tertions are introduced and their properties are studied. The relationship between tertions, complex numbers are quaternions are discussed.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"109 5","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139146970","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 a modification of $underline{Set}(n)$ 关于 $underline{Set}(n)$ 的修改
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-12-01 DOI: 10.7546/nntdm.2023.29.4.813-819
K. Atanassov, József Sándor
{"title":"On a modification of $underline{Set}(n)$","authors":"K. Atanassov, József Sándor","doi":"10.7546/nntdm.2023.29.4.813-819","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.813-819","url":null,"abstract":"A modification of the set $underline{rm Set}(n)$ for a fixed natural number $n$ is introduced in the form: $underline{rm Set}(n, f)$, where $f$ is an arithmetic function. The sets $underline{rm Set}(n,varphi), underline{rm Set}(n,psi), underline{rm Set}(n,sigma)$ are discussed, where $varphi, psi$ and $sigma$ are Euler's function, Dedekind's function and the sum of the positive divisors of $n$, respectively.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"220 4","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138621081","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 t-Fibonacci sequences in the 2-generator p-groups of nilpotency class 2 零能级 2 的 2 个发电机 p 群中的 t-Fibonacci 序列
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-12-01 DOI: 10.7546/nntdm.2023.29.4.827-841
E. Mehraban, Ö. Deveci, E. Hıncal
{"title":"The t-Fibonacci sequences in the 2-generator p-groups of nilpotency class 2","authors":"E. Mehraban, Ö. Deveci, E. Hıncal","doi":"10.7546/nntdm.2023.29.4.827-841","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.827-841","url":null,"abstract":"In this paper, we consider the 2-generator p-groups of nilpotency class 2. We will discuss the lengths of the periods of the t-Fibonacci sequences in these groups.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"125 6","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139191898","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 Wiener, hyper-Wiener, Harary and SK indices of the P(Z_{p^k·q^r}) power graph P(Z_{p^k-q^r})幂图的维纳指数、超维纳指数、哈拉里指数和 SK 指数
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-11-30 DOI: 10.7546/nntdm.2023.29.4.794-803
Volkan Aşkin
{"title":"The Wiener, hyper-Wiener, Harary and SK indices of the P(Z_{p^k·q^r}) power graph","authors":"Volkan Aşkin","doi":"10.7546/nntdm.2023.29.4.794-803","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.794-803","url":null,"abstract":"The undirected $P(Z_n)$ power graph of a finite group of $Z_n$ is a connected graph, the set of vertices of which is $Z_n$. Here $langle u, vrangle in P(Z_n)$ are two diverse adjacent vertices if and only if $u ne v$ and $langle v rangle subseteq langle u rangle$ or $langle u rangle subseteq langle v rangle$. We will shortly name the undirected $P(Z_n)$ power graph as the power graph $P(Z_n)$. The Wiener, hyper-Wiener, Harary and SK indices of the $P(Z_n)$ power graph are in order as follows $$frac{1}{2}underset{left{ u,v right}subseteq Vleft( G right)}{mathop sum },dleft( u,v right), frac{1}{2}underset{left{ u,v right}subseteq Vleft( G right)}{mathop sum },dleft( u,v right)+frac{1}{2}underset{left{ u,v right}subseteq Vleft( G right)}{mathop sum },{{d}^{2}}left(u,v right),$$ $$underset{left{ u,v right}subseteq Vleft( G right)}{mathop sum },frac{1}{dleft(u,v right)} mbox{ and } frac{1}{2}underset{uvin Eleft( G right)}{mathop sum },left( {{d}_{u}}+{{d}_{v}} right).$$ In this article we focus more on the indices of $P(Z_n)$ power graph by Wiener, hyper-Wiener, Harary and SK the definition of the power graph is presented and the results and theorems which we need in our discussion are provided in the introduction. Finally, the main point of the article is that we calculate the Wiener, hyper-Wiener, Harary and SK indices of the power graph $P(Z_n)$ corresponding to the vertex $n = p^k cdot q^r$. These are as follows: $p, q$ are distinct primes and $k, r$ are nonnegative integers.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"67 5","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139205678","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 generalized hyperharmonic numbers of order r, H_{n,m}^{r} (sigma) 关于r阶广义超谐波数 H_{n,m}^{r} (sigma)
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-11-30 DOI: 10.7546/nntdm.2023.29.4.804-812
S. Koparal, N. Ömür, Laid Elkhiri
{"title":"On generalized hyperharmonic numbers of order r, H_{n,m}^{r} (sigma)","authors":"S. Koparal, N. Ömür, Laid Elkhiri","doi":"10.7546/nntdm.2023.29.4.804-812","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.804-812","url":null,"abstract":"In this paper, we define generalized hyperharmonic numbers of order $r, H_{n,m}^{r}left( sigma right) ,$ for $min mathbb{Z}^{+}$ and give some applications by using generating functions of these numbers. For example, for $n, r, sin mathbb{Z}^{+}$ such that $1leq sleq r,$ begin{equation*} sumlimits_{k=1}^{n}binom{n-k+s-1}{s-1}H_{k,m}^{r-s}left( sigma right) =H_{n,m}^{r}left( sigma right), end{equation*} and begin{equation*} sum_{k=1}^{n}sum_{i=1}^{k}frac{H_{k-i,m}^{r+1}left( sigma right) D_{r}(k-i+r)}{(n-k)!left( k-i+rright) !}=H_{n,m}^{2r+2}(sigma ), end{equation*} where $D_{r}(n)$ is an $r$-derangement number.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"15 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139200086","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
New Fibonacci-type pulsated sequences 新的斐波纳契型脉动序列
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-11-30 DOI: 10.7546/nntdm.2023.29.4.789-793
Lilija Atanassova, Velin Andonov
{"title":"New Fibonacci-type pulsated sequences","authors":"Lilija Atanassova, Velin Andonov","doi":"10.7546/nntdm.2023.29.4.789-793","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.789-793","url":null,"abstract":"A new Fibonacci-type sequence from pulsated type is introduced. The explicit form of its members is given.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"120 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139200594","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
Notes on generalized and extended Leonardo numbers 关于广义莱昂纳多数和扩展莱昂纳多数的说明
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-11-27 DOI: 10.7546/nntdm.2023.29.4.752-773
Anthony G. Shannon, P. Shiue, Shen C. Huang
{"title":"Notes on generalized and extended Leonardo numbers","authors":"Anthony G. Shannon, P. Shiue, Shen C. Huang","doi":"10.7546/nntdm.2023.29.4.752-773","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.752-773","url":null,"abstract":"This paper both extends and generalizes recently published properties which have been developed by many authors for elements of the Leonardo sequence in the context of second-order recursive sequences. It does this by considering the difference equation properties of the homogeneous Fibonacci sequence and the non-homogeneous properties of their Leonardo sequence counterparts. This produces a number of new identities associated with a generalized Leonardo sequence and its associated algorithm, as well as some combinatorial results which lead into elegant properties of hyper-Fibonacci numbers in contrast to their ordinary Fibonacci number analogues, and as a convolution of Fibonacci and Leonardo numbers.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"98 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139230156","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
n-Rooks and n-queens problem on planar and modular chessboards with hexagonal cells 带六边形单元的平面和模块棋盘上的 n 车和 n 王问题
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-11-27 DOI: 10.7546/nntdm.2023.29.4.774-788
Eduard C. Taganap, Rainier D. Almuete
{"title":"n-Rooks and n-queens problem on planar and modular chessboards with hexagonal cells","authors":"Eduard C. Taganap, Rainier D. Almuete","doi":"10.7546/nntdm.2023.29.4.774-788","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.774-788","url":null,"abstract":"We show the existence of solutions to the n-rooks problem and n-queens problem on chessboards with hexagonal cells, problems equivalent to certain three and six direction riders on ordinary chessboards. Translating the problems into graph theory problems, we determine the independence number (maximum size of independent set) of rooks graph and queens graph. We consider the $n times n$ planar diamond-shaped H_n with hexagonal cells, and the board $H_n$ as a flat torus $T_n$. Here, a rook can execute moves on lines perpendicular to the six sides of the cell it is placed, and a queen can execute moves on those lines together with lines through the six corners of the cell it is placed.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139234926","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 2-adic valuation of the general degree-2 polynomial in 2 variables 2 变量中一般 2 级多项式的 2-adic 估值
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-11-26 DOI: 10.7546/nntdm.2023.29.4.737-751
Shubham
{"title":"The 2-adic valuation of the general degree-2 polynomial in 2 variables","authors":"Shubham","doi":"10.7546/nntdm.2023.29.4.737-751","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.737-751","url":null,"abstract":"We define the $p$-adic valuation tree of a polynomial $f(x,y)$ $in$ $mathbb{Z}[x,y]$ by which we can find its $p$-adic valuation at any point. This work includes diverse $2$-adic valuation trees of certain degree-two polynomials in two variables. Among these, the $2$-adic valuation tree of $x^2+y^2$ is most interesting. We use the observations from these trees to study the $2$-adic valuation tree of the general degree-two polynomial in $2$ variables. We also study the $2$-adic valuation tree of the polynomial $x^2y+5$.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139235549","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
Lower bounds on expressions dependent on functions φ(n), ψ(n) and σ(n) 取决于函数 φ(n)、ψ(n) 和 σ(n) 的表达式的下界
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-11-21 DOI: 10.7546/nntdm.2023.29.4.713-716
Stoyan Dimitrov
{"title":"Lower bounds on expressions dependent on functions φ(n), ψ(n) and σ(n)","authors":"Stoyan Dimitrov","doi":"10.7546/nntdm.2023.29.4.713-716","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.713-716","url":null,"abstract":"The inequalities varphi^2(n)+psi^2(n)+sigma^2(n) geq 3n^2+2n+3 , varphi(n)psi(n)+varphi(n)sigma(n)+sigma(n)psi(n) geq 3n^2+2n-1 connecting varphi(n), psi(n) and sigma(n)-functions are formulated and proved.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"11 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139251422","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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