**Volume 24** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Happy φ(164)-th birthday, Prof. Tony Shannon!**

*Editorial. Pages 1—2*

Krassimir T. Atanassov and József Sándor

Editorial (PDF, 250 Kb)

**On a new arithmetic function**

*Original research paper. Pages 3—10*

Krassimir T. Atanassov and József Sándor

Full paper (PDF, 177 Kb) | Abstract

**An identical equation for arithmetic functions of several variables and applications**

*Original research paper. Pages 11—17*

Vichian Laohakosol and Pinthira Tangsupphathawat

Full paper (PDF, 180 Kb) | Abstract

**Revisiting some old results on odd perfect numbers**

*Original research paper. Pages 18—25*

Jose Arnaldo Bebita Dris and Doli-Jane Uvales Tejada

Full paper (PDF, 177 Kb) | Abstract

**The Redheffer numbers and their applications**

*Original research paper. Pages 26—37*

Ömür Deveci and Anthony G. Shannon

Full paper (PDF, 142 Kb) | Abstract

*m*. Furthermore, we define the Redheffer orbits and the basic Redheffer orbits of 2-generator and 3-generator groups, then we examine the lengths of the periods of these orbits. Finally, we obtain the Redheffer lengths and the basic Redheffer lengths of some special finite groups as applications of Redheffer orbits and the basic Redheffer orbits.

**Results on generalized negabent functions**

*Original research paper. Pages 38—44*

Rashmeet Kaur and Deepmala Sharma

Full paper (PDF, 166 Kb) | Abstract

_{2}

^{n}with values in ℤ

_{8}and ℤ

_{16}. Furthermore, we propose several constructions of generalized negabent functions.

**A new symmetric endomorphism operator for some generalizations of certain generating functions**

*Original research paper. Pages 45—58*

Ali Boussayoud, Abdelhamid Abderrezzak and Serkan Araci

Full paper (PDF, 207 Kb) | Abstract

*k*-Jacobsthal numbers and Tchebychev polynomials of the first and second kind.

**Simple applications of continued fractions and an elementary result on Heron’s algorithm**

*Original research paper. Pages 59—69*

Antonino Leonardis

Full paper (PDF, 904 Kb) | Abstract

*AMS special session on Continued Fraction*during the

*Joint Mathematical Meetings 2017, Atlanta GA*. The first part is more introductory/educational, explaining the importance of matricial and Diophantine methods in the topic of continued fractions. We will begin this part discussing geometrical illusions which can arise from properties of continued fractions and associated matrices, proving thoroughly the mathematical reasons of this fact. After this, we will deal with the Pythagorean problem of the right-angled isosceles triangles finding all solutions to the simple Diophantine equation

*l*

^{2}+ (

*l*+ 1)

^{2}=

*d*

^{2}, which will give a “Pseudo-Pythagorean” triangle. In the second part, recalling all the methods introduced in the first one, we will prove a theorem (the main result), which relates continued fractions with Heron’s algorithm, giving some examples. This theorem is proved in a complete form, which considers all possibilities and its vice-versa, unlike all other minor results that can be found in the literature (see 〚2, 3⟧).

**Generalized dual Fibonacci quaternions with dual coefficient**

*Original research paper. Pages 70—85*

Fügen Torunbalcı Aydın

Full paper (PDF, 226 Kb) | Abstract

**The structure of prime sums**

*Original research paper. Pages 86—91*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 114 Kb) | Abstract

**Some combinatorial identities via Stirling transform**

*Original research paper. Pages 92—98*

Fouad Bounebirat, Diffalah Laissaoui and Mourad Rahmani

Full paper (PDF, 191 Kb) | Abstract

**Explicit expression for symmetric identities of w-Catalan–Daehee polynomials**

*Original research paper. Pages 99—111*

Taekyun Kim, Seog-Hoon Rim, Dmitry V. Dolgy and Sung-Soo Pyo

Full paper (PDF, 209 Kb) | Abstract

*w*-Catalan–Daehee polynomials and investigate some properties for those polynomials. In addition, we give explicit expression for the symmetric identities of the

*w*-Catalan–Daehee polynomials which are derived from p-adic invariant integral on ℤ

_{p}.

**On the Diophantine equation L_{n} − L_{m} = 3 • 2^{a}**

*Original research paper. Pages 112—119*

Zafer Şiar and Refik Keskin

Full paper (PDF, 191 Kb) | Abstract

*m*,

*n*and

*a*. It is shown that solutions of the equation

*L*−

_{n}*L*= 3 • 2

_{m}^{a}are given by

*L*

_{11}−

*L*

_{4}= 199 − 7 = 3 • 2

^{6},

*L*

_{4}−

*L*

_{3}= 7 − 4 = 3 • 2

^{0},

*L*

_{4}−

*L*

_{1}= 7 − 1 = 3 • 2,

*L*

_{3}−

*L*

_{1}= 4 − 1 = 3 • 2

^{0}. In order to prove our result, we use lower bounds for linear forms in logarithms and a version of the Baker–Davenport reduction method in Diophantine approximation.

**On the convergence of second-order recurrence series**

*Original research paper. Pages 120—127*

Bijan Kumar Patel and Prasanta Kumar Ray

Full paper (PDF, 150 Kb) | Abstract

**Valuations, arithmetic progressions, and prime numbers**

*Original research paper. Pages 128—132*

Shin-ichiro Seki

Full paper (PDF, 156 Kb) | Abstract

**Canonical matrices with entries integers modulo p**

*Original research paper. Pages 133—143*

Krasimir Yordzhev

Full paper (PDF, 173 Kb) | Abstract

*n*×

*m*matrices with entries in the set = {0, 1, …

*p*− 1}. In each element of the factor-set generated by this relation, we define the concept of canonical matrix, namely the minimal element with respect to the lexicographic order. We have found a necessary and sufficient condition for an arbitrary matrix with entries in the set to be canonical. For this purpose, the matrices are uniquely represented by ordered

*n*-tuples of integers.

**Extremal chemical trees of the first reverse Zagreb beta index**

*Original research paper. Pages 144—148*

Süleyman Ediz and Mesut Semiz

Full paper (PDF, 133 Kb) | Abstract

*v*of a simple connected graph

*G*defined as

*c*= Δ −

_{v}*d*+ 1, where Δ denotes the largest of all degrees of vertices of

_{v}*G*and

*d*denotes the number of edges incident to

_{v}*v*. The first reverse Zagreb beta index of a simple connected graph

*G*defined as

*CM*

^{β}_{1}(

*G*) = Σ

_{uv∈E(G)}(

*c*+

_{u}*c*). In this paper, we characterized maximum chemical trees with respect to the first reverse Zagreb beta index.

_{v}**Editorial note**

*Pages 149—150*

Editorial (PDF, 35 Kb)

*This issue of the International Journal “Notes on Number Theory and Discrete Mathematics” is published with the financial support of the Bulgarian National Science Fund, Grant Ref. No. DNP-06-38/2017.*