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Some new relations between T(a1,a2,a3,a4,a5;n) and N(a1,a2,a3,a4,a5;n) T(a1,a2,a3,a4,a5;n)与n (a1,a2,a3,a4,a5;n)的新关系
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-04-25 DOI: 10.7546/nntdm.2023.29.2.216-225
Vandna Vandna, Mandeep Kaur
{"title":"Some new relations between T(a1,a2,a3,a4,a5;n) and N(a1,a2,a3,a4,a5;n)","authors":"Vandna Vandna, Mandeep Kaur","doi":"10.7546/nntdm.2023.29.2.216-225","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.216-225","url":null,"abstract":"Let $N(a_1,a_2,a_3,a_4,a_5;n)$ and $T(a_1,a_2,a_3,a_4,a_5;n)$ count the representations of $n$ as $a_1x_1^2+a_2x_2^2+a_3x_3^2+a_4x_4^2+a_5x_5^2$ and $a_1X_1(X_1+1)/2+a_2X_2(X_2+1)/2+a_3X_3(X_3+1)/2+a_4X_4(X_4+1)/2+a_5X_5(X_5+1)/2$, respectively, where $a_1,a_2,a_3,a_4,a_5$ are positive integers, $x_1,x_2,x_3,x_4,x_5$ are integers and $n,X_1,X_2,X_3,X_4,X_5$ are nonnegative integers. In this paper, we establish some new relations between $N(a_1,a_2,a_3,a_4,a_5;n)$ and $T(a_1,a_2,a_3,a_4,a_5;n)$. Also, we prove that $T(a_1,a_2,a_3,a_4,a_5;n)$ is a linear combination of $N(a_1,a_2,a_3,a_4,a_5;m)$ and $N(a_1,a_2,a_3,a_4,a_5;m/4)$, where $m=8n+a_1+a_2+a_3+a_4+a_5$, for various values of $a_1,a_2,a_3,$ $a_4,a_5$.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45647118","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
Some identities involving Chebyshev polynomials, Fibonacci polynomials and their derivatives 涉及切比雪夫多项式、斐波那契多项式及其导数的一些恒等式
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-04-16 DOI: 10.7546/nntdm.2023.29.2.204-215
J. Kishore, V. Verma
{"title":"Some identities involving Chebyshev polynomials, Fibonacci polynomials and their derivatives","authors":"J. Kishore, V. Verma","doi":"10.7546/nntdm.2023.29.2.204-215","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.204-215","url":null,"abstract":"In this paper, we will derive the explicit formulae for Chebyshev polynomials of the third and fourth kind with odd and even indices using the combinatorial method. Similar results are also deduced for their r-th derivatives. Finally, some identities involving Chebyshev polynomials of the third and fourth kind with even and odd indices and Fibonacci polynomials with negative indices are obtained.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48695652","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: 2
A note on telephone numbers and their matrix generators 关于电话号码及其矩阵生成器的说明
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-04-12 DOI: 10.7546/nntdm.2023.29.2.195-203
F. R. Alves, R. Vieira, P. Catarino
{"title":"A note on telephone numbers and their matrix generators","authors":"F. R. Alves, R. Vieira, P. Catarino","doi":"10.7546/nntdm.2023.29.2.195-203","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.195-203","url":null,"abstract":"In the present work, we indicate some matrix properties that allow us to determine new relationships involving telephone numbers and some products that allow us to obtain telephone terms, based on second-order matrices.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48963802","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
Quotients of arithmetical functions under the Dirichlet convolution 狄利克雷卷积下算术函数的商
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-04-03 DOI: 10.7546/nntdm.2023.29.2.185-194
P. Haukkanen
{"title":"Quotients of arithmetical functions under the Dirichlet convolution","authors":"P. Haukkanen","doi":"10.7546/nntdm.2023.29.2.185-194","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.185-194","url":null,"abstract":"We study existence of a solution of the arithmetical equation $fast h = g$ in $f,$ where $fast h$ is the Dirichlet convolution of arithmetical functions $f$ and $h,$ and derive an explicit expression for the solution. As applications we obtain expressions for the Möbius function $mu$ and the so-called totients. As applications we also present our results on the arithmetical equation $fast h = g$ in the language of Cauchy convolution and further deconvolution in discrete linear systems.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43395865","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 generalized computation procedure for Ramanujan-type identities and cubic Shevelev sum ramanujan型恒等式和三次舍夫列夫和的一种广义计算方法
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-03-01 DOI: 10.7546/nntdm.2023.29.1.98-129
P. Shiue, A. Shannon, Shen C. Huang, Jorge E. Reyes
{"title":"A generalized computation procedure for Ramanujan-type identities and cubic Shevelev sum","authors":"P. Shiue, A. Shannon, Shen C. Huang, Jorge E. Reyes","doi":"10.7546/nntdm.2023.29.1.98-129","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.98-129","url":null,"abstract":"A generalized Computation procedure for construction of the Ramanujan-type from a given general cubic equation and a cosine Ramanujan-type identity is developed from detailed analyses of the properties of Ramanujan-type cubic equations. Examples are provided together with cubic Shevelev sums.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47476002","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
Hybrid hyper-Fibonacci and hyper-Lucas numbers 混合超斐波那契数和超卢卡斯数
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-03-01 DOI: 10.7546/nntdm.2023.29.1.154-170
Yasemin Alp
{"title":"Hybrid hyper-Fibonacci and hyper-Lucas numbers","authors":"Yasemin Alp","doi":"10.7546/nntdm.2023.29.1.154-170","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.154-170","url":null,"abstract":"Different number systems have been studied lately. Recently, many researchers have considered the hybrid numbers which are generalization of the complex, hyperbolic and dual number systems. In this paper, we define the hybrid hyper-Fibonacci and hyper-Lucas numbers. Furthermore, we obtain some algebraic properties of these numbers such as the recurrence relations, the generating functions, the Binet’s formulas, the summation formulas, the Catalan’s identity, the Cassini’s identity and the d’Ocagne’s identity.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48124108","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: 2
Objects generated by an arbitrary natural number. Part 3: Standard modal-topological aspect 由任意自然数生成的对象。第3部分：标准模态拓扑方面
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-03-01 DOI: 10.7546/nntdm.2023.29.1.171-180
K. Atanassov
{"title":"Objects generated by an arbitrary natural number. Part 3: Standard modal-topological aspect","authors":"K. Atanassov","doi":"10.7546/nntdm.2023.29.1.171-180","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.171-180","url":null,"abstract":"The set $underline{SET}(n)$ generated by an arbitrary natural number $n$, was defined in [3]. There, and in [4], some arithmetic functions and arithmetic operators of a modal type are defined over the elements of $underline{SET}(n)$. Here, over the elements of $underline{SET}(n)$ arithmetic operators of a topological type are defined and some of their basic properties are studied. Perspectives for future research are discussed.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46666891","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 equations and inequalities involving the arithmetical functions φ(n) and d(n) – II 有关算术函数φ(n)和d(n) - II的若干方程和不等式
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-03-01 DOI: 10.7546/nntdm.2023.29.1.130-136
József Sándor
{"title":"On certain equations and inequalities involving the arithmetical functions φ(n) and d(n) – II","authors":"József Sándor","doi":"10.7546/nntdm.2023.29.1.130-136","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.130-136","url":null,"abstract":"In papers [3] and [5] we have studied certain equations and inequalities involving the arithmetical functions varphi(n) and d(n). In this paper we will consider some other equations. Some open problems will be stated, too.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135185321","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
Corrigendum to: “Some modular considerations regarding odd perfect numbers – Part II” [Notes on Number Theory and Discrete Mathematics, 2020, Vol. 26, No. 3, 8–24] “关于奇完全数的一些模块化考虑-第二部分”的勘误表[数论与离散数学注释，2020,Vol. 26, No. 3, 8-24]
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-03-01 DOI: 10.7546/nntdm.2023.29.1.181-184
J. A. Dris, Immanuel Tobias San Diego
{"title":"Corrigendum to: “Some modular considerations regarding odd perfect numbers – Part II” [Notes on Number Theory and Discrete Mathematics, 2020, Vol. 26, No. 3, 8–24]","authors":"J. A. Dris, Immanuel Tobias San Diego","doi":"10.7546/nntdm.2023.29.1.181-184","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.181-184","url":null,"abstract":"In [2], the authors proposed a theorem which they recently found out to contradict Chen and Luo’s results [1]. In the present paper, we provide the correct form of this theorem.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48220599","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 edge irregularity strength of firefly graph 萤火虫图边缘不规则强度的一个注记
IF 0.3
Notes on Number Theory and Discrete Mathematics Pub Date : 2023-03-01 DOI: 10.7546/nntdm.2023.29.1.147-153
Umme Salma, H. M. Nagesh, D. Prahlad
{"title":"A note on edge irregularity strength of firefly graph","authors":"Umme Salma, H. M. Nagesh, D. Prahlad","doi":"10.7546/nntdm.2023.29.1.147-153","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.147-153","url":null,"abstract":"Let $G$ be a simple graph. A vertex labeling $psi:V(G) rightarrow {1, 2,ldots,alpha}$ is called $alpha$-labeling. For an edge $uv in G$, the weight of $uv$, written $z_{psi}(uv)$, is the sum of the labels of $u$ and $v$, i.e., $z_{psi}(uv)=psi(u)+psi(v)$. A vertex $alpha$-labeling is said to be an edge irregular $alpha$-labeling of $G$ if for every two distinct edges $a$ and $b$, $z_{psi}(a) neq z_{psi}(b)$. The minimum $alpha$ for which the graph $G$ contains an edge irregular $alpha$-labeling is known as the edge irregularity strength of $G$ and is denoted by $es(G)$. In this paper, we find the exact value of edge irregularity strength of different cases of firefly graph $F_{s,t,n-2s-2t-1}$ for any $s geq 1, t geq 1, n-2s-2t-1 geq 1 $.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43349816","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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