**Volume 26** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Bi-unitary multiperfect numbers, II**

*Original research paper. Pages 1–26*

Pentti Haukkanen and Varanasi Sitaramaiah

Full paper (PDF, 258 Kb) | Abstract

Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers. The present paper is Part II in a series of papers on even bi-unitary multiperfect numbers. In the first part we found all bi-unitary triperfect numbers of the form , where and is odd; the only one being . In this second part we find all bi-unitary triperfect numbers in the cases and . For the only one is , and for they are , , and

**Some modular considerations regarding odd perfect numbers**

*Original research paper. Pages 27–33*

Jose Arnaldo Bebita Dris and Immanuel Tobias San Diego

Full paper (PDF, 143 Kb) | Abstract

**Restrictive factor and extension factor**

*Original research paper. Pages 34–46*

József Sándor and Krassimir T. Atanassov

Full paper (PDF, 214 Kb) | Abstract

**On multiplicative order of elements in finite fields based on cyclotomic polynomials**

*Original research paper. Pages 47–52*

Roman Popovych

Full paper (PDF, 97 Kb) | Abstract

**Number of tuples with a given least common multiple**

*Original research paper. Pages 53–60*

K. Siddharth Choudary and A. Satyanarayana Reddy

Full paper (PDF, 170 Kb) | Abstract

**On the symmetrical second order hyperbolic quaternions sequences**

*Original research paper. Pages 61–70*

Sure Köme and Cahit Köme

Full paper (PDF, 164 Kb) | Abstract

**Higher-order identities for balancing numbers**

*Original research paper. Pages 71–84*

Takao Komatsu

Full paper (PDF, 186 Kb) | Abstract

.

This type can be generalized, so that is a special case of the number , where () with and .

**Infinite series containing generalized harmonic functions**

*Original research paper. Pages 85–104*

Kwang-Wu Chen and Yi-Hsuan Chen

Full paper (PDF, 287 Kb) | Abstract

**The linear combination of two polygonal numbers is a perfect square**

*Original research paper. Pages 105–115*

Mei Jiang and Yangcheng Li

Full paper (PDF, 200 Kb) | Abstract

then the Diophantine equation has infinitely many positive integer solutions . Moreover, we give conditions about such that the Diophantine equation has infinitely many positive integer solutions .

**An alternative proof of Nyblom’s results and a generalisation**

*Original research paper. Pages 116–126*

A. David Christopher

Full paper (PDF, 191 Kb) | Abstract

and to be the number of ways can be expressed as a difference of two elements from the sequence . Nyblom found closed expressions for and in terms of some restricted number-of-divisors functions. Here we re-establish these two results of Nyblom in a relatively simple way. Along with the other interpretations for , an expression for is presented in terms of restricted form of and . Also we consider another function due to Nyblom, denoted , which counts the number of partitions of with parts in arithmetic progression having common difference . Nyblom and Evan found a simple expression for and put in terms of a divisor-counting functions when . Here we re-establish Nyblom’s expression for , and find equinumerous expressions for when . Finally, we present the following generalised version of : given a set of positive integers say, , we denote by , the number of ways can be written as a difference of two elements from the set . And we express in terms of partition enumerations when some restrictions are imposed upon the elements of . We close with the hint that, boundedness of together with the divergence of disproves Erdős arithmetic progression conjecture.

**Multifarious results for q-Hermite-based Frobenius-type Eulerian polynomials**

*Original research paper. Pages 127–141*

Waseem Ahmad Khan, Idrees Ahmad Khan, Mehmet Acikgoz and Ugur Duran

Full paper (PDF, 220 Kb) | Abstract

*q*-Hermite-based Frobenius-type Eulerian polynomials is introduced by means of generating function and series representation. Several fundamental formulas and recurrence relations for these polynomials are derived via different generating methods. Furthermore, diverse correlations including the

*q*-Apostol-Bernoulli polynomials, the

*q*-Apostol-Euler poynoomials, the

*q*-Apostol-Genocchi polynomials and the

*q*-Stirling numbers of the second kind are also established by means of the their generating functions.

**Saalschütz’ theorem and Rising binomial coefficients – Type 2**

*Original research paper. Pages 142–147*

A. G. Shannon

Full paper (PDF, 98 Kb) | Abstract

**A new explicit formula for the Bernoulli numbers in terms of the Stirling numbers of the second kind**

*Original research paper. Pages 148–151*

Sumit Kumar Jha

Full paper (PDF, 129 Kb) | Abstract

To the best of our knowledge, the formula is new.

**Infinite sums associated with certain Lucas sequences**

*Original research paper. Pages 152–158*

S. G. Rayaguru and G. K. Panda

Full paper (PDF, 169 Kb) | Abstract

**Infinite series involving Fibonacci numbers and the Riemann zeta function**

*Original research paper. Pages 159–166*

Robert Frontczak

Full paper (PDF, 185 Kb) | Abstract

**Circular-hyperbolic Fibonacci quaternions**

*Original research paper. Pages 167–176*

Fügen Torunbalcı Aydın

Full paper (PDF, 194 Kb) | Abstract

**On r-circulant matrices with Horadam numbers having arithmetic indices**

*Original research paper. Pages 177–197*

Aldous Cesar F. Bueno

Full paper (PDF, 238 Kb) | Abstract

*r*-circulant matrix whose entries are Horadam numbers having arithmetic indices. We then solve for the eigenvalues, determinant, Euclidean norm and spectral norm of the matrix. Lastly, we present some special cases and some results on identities and divisibility.

**On generalized Fibonacci quadratics**

*Original research paper. Pages 198–204*

Neşe Ömür and Zehra Betül Gür

Full paper (PDF, 193 Kb) | Abstract

Fibonacci quadratics and give solutions of them under certain conditions.

For example, for odd number under condition , the equation

has rational roots.

**On generalized order- k modified Pell and Pell–Lucas numbers in terms of Fibonacci and Lucas numbers**

*Original research paper. Pages 205–212*

Ahmet Daşdemir

Full paper (PDF, 345 Kb) | Abstract

*k*Pell–Lucas and Modified Pell numbers can be expressed in terms of the well-known Fibonacci numbers. Certain

*n*-square Hessenberg matrices with permanents equal to the Fibonacci numbers are defined. These Hessenberg matrices are then extended to super-diagonal (0,1,2)-matrices. In particular, the permanents of the super-diagonal matrices are shown to equal the components of the generalized order-

*k*Pell–Lucas and Modified Pell numbers, and also their sums. In addition, two computer algorithms regarding our results are composed.

**An identity for vertically aligned entries in Pascal’s triangle**

*Original research paper. Pages 213–221*

Heidi Goodson

Full paper (PDF, 156 Kb) | Abstract

*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.*