**Volume 23** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**On a curious biconditional involving the divisors of odd perfect numbers**

*Original research paper. Pages 1—13*

Jose Arnaldo B. Dris

Full paper (PDF, 177 Kb) | Abstract

*q*<

^{k}*n*holds, where

*q*

^{k}n^{2}is an odd perfect number with Euler prime

*q*. We then show that this biconditional holds unconditionally. Lastly, we prove that the inequality

*q*<

*n*holds unconditionally.

**Short remark on a special numerical sequence**

*Original research paper. Pages 14—17*

Krassimir T. Atanassov

Full paper (PDF, 152 Kb) | Abstract

*G*= {2

^{2}3

^{3}…

*p*}

_{n}^{pn}_{n ≥ 1}is discussed and some of its properties are studied.

**A note on bounds for the Neuman–Sándor mean using power and identric means**

*Original research paper. Pages 18—21*

József Sándor

Full paper (PDF, 127 Kb) | Abstract

**Prime triples p_{1}, p_{2}, p_{3} in arithmetic progressions such that p_{1} = x^{2} + y^{2} + 1, p_{3} = [n^{c}]**

*Original research paper. Pages 22—33*

S. I. Dimitrov

Full paper (PDF, 225 Kb) | Abstract

*p*

_{1},

*p*

_{2},

*p*

_{3}= 2

*p*

_{2}−

*p*

_{1}such that

*p*

_{1}=

*x*

^{2}+

*y*

^{2}+ 1,

*p*

_{3}=

*n*.

^{c}**Sum of dilates of two sets**

*Original research paper. Pages 34—41*

Raj Kumar Mistri

Full paper (PDF, 151Kb) | Abstract

*A*⊆

*Z*and

*B*⊆

*Z*be nonempty finite sets and let

*r*be a nonzero integer. The sumof dilates of

*A*and

*B*is defined as

*A*+

*r · B*:= {

*a*+

*rb*:

*a*∈

*A*and

*b*∈

*B*}. Finding nontriviallower bound for the sum of dilates is an important problem in additive combinatorics and it hasapplications in sum-product problems. In case of

*A*=

*B*, a recent result of Freiman et al. states that if

*r*≥ 3, then |

*A*+

*r*

*· A|*≥ 4|

*A*| – 4. We generalize this result for the sum of dilates

*A*+

*r*for two sets

*·*B*A*and

*B*, where

*r*is an integer with |

*r*| ≥ 3.

**On limits and formulae where functions of slow increase appear**

*Original research paper. Pages 42—51*

Rafael Jakimczuk

Full paper (PDF, 168 Kb) | Abstract

*A*be a strictly increasing sequence of positive integers such that

_{n}*A*∼

_{n}*n*(

^{s}f*n*), where

*f*(

*x*) is a function of slow increase and

*s*is a positive real number. In this article we obtain some limits and asymptotic formulae where appear functions of slow increase. As example, we apply the obtained results to the sequence of numbers with exactly

*k*prime factors in their prime factorization, in particular to the sequence of prime numbers (

*k*= 1).

**Two applications of the Hadamard integral inequality**

*Original research paper. Pages 52—55*

József Sándor

Full paper (PDF, 127 Kb) | Abstract

*x*/

*x*).

**Some properties of the bi-periodic Horadam sequences**

*Original research paper. Pages 56—65*

Elif Tan

Full paper (PDF, 168 Kb) | Abstract

**Generalized dual Pell quaternions**

*Original research paper. Pages 66—84*

Fügen Torunbalcı Aydın, Kevser Köklü and Salim Yüce

Full paper (PDF, 228 Kb) | Abstract

**Some variations on Fibonacci matrix graphs**

*Original research paper. Pages 85—93*

Anthony G. Shannon and Ömür Deveci

Full paper (PDF, 105 Kb) | Abstract