**Volume 26** ▶ Number 1 ▷ Number 2

**Riemann zeta function and arithmetic progression of higher order**

*Original research paper. Pages 1—7*

Hamilton Brito, Éder Furtado and Fernando Matos

Full paper (PDF, 567 Kb) | Abstract

**Odd and even repetition sequences of independent domination number**

*Original research paper. Pages 8—20*

Leomarich F. Casinillo

Full paper (PDF, 659 Kb) | Abstract

_{n}}

_{n=1}

^{∞}be a sequence of paths. The odd repetition sequence denoted by {𝜌

_{𝑘}

^{𝑜}:𝑘∈ℕ} is a sequence of natural numbers in which odd numbers are repeated once and defined by {𝜌

_{𝑘}

^{𝑜}}={1,1,2,3,3,4,5,5,…}={𝑖(𝑃

_{𝑛})} where 𝑛=2𝑘−1. The even repetition sequence denoted by {𝜌

_{𝑘}

^{𝑒}:𝑘∈ℕ} is a sequence of natural numbers, in which even numbers are repeated once and defined by {𝜌

_{𝑘}

^{𝑒}}={1,2,2,3,4,4,5,6,6,…}={𝑖(𝑃

_{𝑛})}, where 𝑛 = 2𝑘. In this paper, the explicit formula that shows the values of the element of two sequences {𝜌

_{𝑘}

^{𝑜}}} and {𝜌

_{𝑘}

^{𝑒}} that depends on the subscript 𝑘 were constructed. Also, the formula that relates the partial sum of the elements of the said sequences, which depends on the subscript 𝑘 and order of the sequence of paths, were established. Further, the independent domination number of the triangular grid graph 𝑇

_{𝑚}= (𝑉(𝑇

_{𝑚}), 𝐸(𝑇

_{𝑚})) will be determined using the said sequences and the two sequences will be evaluated in relation to the Fibonacci sequence {𝐹

_{𝑛}} along with the order of the path.

**Square-full numbers with an even number of prime factors**

*Original research paper. Pages 21—30*

Rafael Jakimczuk

Full paper (PDF, 183 Kb) | Abstract

*ω*(

*n*) and Ω(

*n*), where

*n*is an

*s*-full number. For example, we prove that the square-full numbers with Ω(

*n*) even are in greater proportion than the square-full numbers with Ω(

*n*) odd. The methods used are elementary.

**Extension factor: Definition, properties and problems. Part 2**

*Original research paper. Pages 31—39*

Krassimir T. Atanassov and József Sándor

Full paper (PDF, 168 Kb) | Abstract

**A parametrised family of Mordell curves with a rational point of order 3**

*Original research paper. Pages 40—44*

Ajai Choudhry and Arman Shamsi Zargar

Full paper (PDF, 115 Kb) | Abstract

*y*

^{2}=

*x*

^{3}+

*d*is called a Mordell curve. This paper is concerned with Mordell curves for which

*d*=

*k*

^{2};

*k*∈ ℤ;

*k*≠ 1. The point (0,

*k*) on such curves is of order 3 and the torsion subgroup of the group of rational points on such Mordell curves is necessarily ℤ/3ℤ. We obtain a parametrised family of Mordell curves

*y*

^{2}=

*x*

^{3}+

*k*

^{2}such that the rank of each member of the family is at least 3. Some elliptic curves of the family have ranks 4 and 5.

*t*-cobalancing numbers and *t*-cobalancers

*Original research paper. Pages 45—58*

Ahmet Tekcan and Alper Erdem

Full paper (PDF, 217 Kb) | Abstract

*t*-cobalancers,

*t*-cobalancing numbers and Lucas

*t*-cobalancing numbers by solving the Pell equation 2

*x*

^{2}−

*y*

^{2}= 2

*t*

^{2}− 1 for some fixed integer

*t*≥ 1.

**Classical pairs in Z_{n}**

*Original research paper. Pages 59—69*

Tekuri Chalapathi, Shaik Sajana and Dasari Bharathi

Full paper (PDF, 716 Kb) | Abstract

*Z*whose least common multiple is zero and concentrate the properties of these pairs. We establish a formula for determining the number of classical pairs in

_{n}*Z*for various values of

_{n}*n*. Further, we present an algorithm for determining all these pairs in

*Z*.

_{n}**A bound of sums with convolutions of Dirichlet characters**

*Original research paper. Pages 70—74*

Teerapat Srichan

Full paper (PDF, 157 Kb) | Abstract

**A new class of q-Hermite-based Apostol-type polynomials and its applications**

*Original research paper. Pages 75—85*

Waseem A. Khan and Divesh Srivastava

Full paper (PDF, 196 Kb) | Abstract

*q*-Hermite based Apostol-type polynomials and to investigate their properties and characteristics. In particular, the generating functions, series expression and explicit and recurrence relations for these polynomials are established. We derive some relationships for

*q*-Hermite based Apostol-type polynomials associated with

*q*-Apostol-type Bernoulli polynomials,

*q*-Apostol-type Euler and

*q*-Apostol-type Genocchi polynomials.

**A note on identities in two variables for a class of monoids**

*Original research paper. Pages 86—92*

Enrique Salcido and Emil Daniel Schwab

Full paper (PDF, 178 Kb) | Abstract

*X*= {

*x*,

*y*}. This note is self-contained and the aim is to describe gradually the identities partition (with three parameters) of the free semigroup

*X*

^{+}for the class of monoids

*B*= {

_{n}*a*,

*b*|

*ba*=

*b*} (

^{n}*n*> 0).

**Bi-unitary multiperfect numbers, I**

*Original research paper. Pages 93—171*

Pentti Haukkanen and Varanasi Sitaramaiah

Full paper (PDF, 432 Kb) | Abstract

*d*of a positive integer

*n*is called a unitary divisor if gcd(

*d*,

*n*/

*d*) = 1; and

*d*is called a bi-unitary divisor of

*n*if the greatest common unitary divisor of

*d*and

*n*/

*d*is unity. The concept of a bi-unitary divisor is due to D. Surynarayana [12]. Let

*σ*

^{∗∗}(

*n*) denote the sum of the bi-unitary divisors of

*n*. A positive integer

*n*is called a bi-unitary perfect number if

*σ*

^{∗∗}(

*n*) = 2

*n*. This concept was introduced by C. R. Wall in 1972 [15], and he proved that there are only three bi-unitary perfect numbers, namely 6, 60 and 90.

In 1987, Peter Hagis [6] introduced the notion of bi-unitary multi

*k*-perfect numbers as solutions

*n*of the equation

*σ*

^{∗∗}(

*n*) =

*kn*. A bi-unitary multi 3-perfect number is called a bi-unitary triperfect number. A bi-unitary multiperfect number means a bi-unitary multi

*k*-perfect number with

*k*≥ 3. Hagis [6] proved that there are no odd bi-unitary multiperfect numbers. We aim to publish a series of papers on bi-unitary multiperfect numbers focusing on multiperfect numbers of the form

*n*= 2

*, where*

^{a}u*u*is odd. In this paper—part I of the series—we investigate bi-unitary triperfect numbers of the form

*n*= 2

*, where 1 ≤*

^{a}u*a*≤ 3. It appears that

*n*= 120 = 2

^{3}15 is the only such number. Hagis [6] found by computer the bi-unitary multiperfect numbers less than 10

^{7}. We have found 31 such numbers up to 8.10

^{10}. The first 13 are due to Hagis. After completing this paper we noticed that further numbers are already listed in The On-Line Encyclopedia of Integer Sequences (sequence A189000 Bi-unitary multiperfect numbers). The numbers listed there have been found by direct computer calculations. Our purpose is to present a mathematical search and treatment of bi-unitary multiperfect numbers.

**Phidias numbers as a basis for Fibonacci analogues**

*Original research paper. Pages 172—178*

P. S. Kosobutskyy

Full paper (PDF, 452 Kb) | Abstract

*p*| = |

*q*| ≠ 1 having properties of the numbers of Phidias

*φ = 0.*61803… and Ф = 1.61803… It is shown that it is also possible to construct a set of sequences possessing the basic properties of the Fibonacci and Lucas sequences.

**On the Generalized Fibonacci-circulant-Hurwitz numbers**

*Original research paper. Pages 179—190*

Ömür Deveci, Zafer Adıgüzel and Taha Doğan

Full paper (PDF, 182 Kb) | Abstract

**On hyper-dual generalized Fibonacci numbers**

*Original research paper. Pages 191—198*

Ne*ş*e Ömür and Sibel Koparal

Full paper (PDF, 163 Kb) | Abstract

**Some binomial-sum identities for the generalized bi-periodic Fibonacci sequences**

*Original research paper. Pages 199—208*

Ho-Hon Leung

Full paper (PDF, 164 Kb) | Abstract

*n*-th term considered is odd or even. In this paper, we investigate the properties of the generalized bi-periodic Fibonacci sequences. It is a generalization of the biperiodic Fibonacci sequences defined by Edson and Yayenie. We derive binomial-sum identities for the generalized bi-periodic Fibonacci sequences by matrix method. Our identities generalize binomial-sum identities derived by Edson and Yayenie for the case of bi-periodic Fibonacci sequences.

**Identities on the product of Jacobsthal-like and Jacobsthal–Lucas numbers**

*Original research paper. Pages 209—215*

Apisit Pakapongpun

Full paper (PDF, 142 Kb) | Abstract

**Some properties of ( p, q)-Fibonacci-like and (p, q)-Lucas numbers**

*Original research paper. Pages 216—224*

Boonyen Thongkam, Khuanphirom Butsuwan and Phonpriya Bunya

Full paper (PDF, 144 Kb) | Abstract

*p, q*)-Fibonacci-like and (

*p, q*)-Lucas numbers. It also reveals some properties on the products of (

*p, q*)-Fibonacci like and (

*p, q*)-Lucas numbers.

**Gaussian binomial coefficients**

*Original research paper. Pages 225—229*

A. G. Shannon

Full paper (PDF, 523 Kb) | Abstract

**Corrigendum to “On the Iyengar–Madhava Rao–Nanjundiah inequality and its hyperbolic version” [Notes on Number Theory and Discrete Mathematics, Vol. 24, 2018, No. 2, 134–139]**

*Corrigendum. Page 230*

József Sándor

Corrigendum (PDF, 270 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. KP-06-NP1-15/2019.*