**Volume 20** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4 ▷ Number 5

**A set of Lucas sequences**

*Original research paper. Pages 1—5*

Krassimir Atanassov

Full paper (PDF, 146 Kb) | Abstract

A new extension of the concept of Fibonacci–like sequences is constructed, related to Lucas sequence. Some of its properties are discussed.

**Fibonacci primes**

*Original research paper. Pages 6—9*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 85 Kb) | Abstract

Fibonacci composites with prime subscripts, *F*_{p}, have factors (*kp *± 1) in which *k* is even. The class of *p* governs the class of *k* in the modular ring *Z*_{5}, and the digit sum of *p*, *F*_{p} and a function of *F*_{p} provide an approximate check on primality.

**Fibonacci primes of special forms**

*Original research paper. Pages 10—19*

Diana Savin

Full paper (PDF, 202 Kb) | Abstract

A study of Fibonacci primes of the form *x*^{2} + *ry*^{2} (where *r* = 1; *r* = prime or *r* = perfect power) is provided.

**A Diophantine system about equal sum of cubes**

*Pages 20—28*

Zhi Ren

Removed paper!

As of April 2016, this paper has been removed from the server!

We were notified that an identical version of the paper has been published in the *Journal of Integer Sequences*, Vol. 16(2013), Article 13.7.8., without the author informing in advance the Editorial Board of the *Notes on Number Theory and Discrete Mathematics*.

Please, accept our apologies for any inconvenience.

**On two Diophantine equations 2***A*^{6} + *B*^{6} = 2*C*^{6} ± *D*^{3}

*Original research paper. Pages 29—34*

Susil Kumar Jena

Full paper (PDF, 138 Kb) | Abstract

We give parametric solutions, and thus show that the two Diophantine equations 2*A*^{6} + *B*^{6} = 2*C*^{6} ± *D*^{3} have infinitely many nontrivial and primitive solutions in positive integers (*A, **B*, *C*, *D*).

**An explicit estimate for the Barban and Vehov weights**

*Original research paper. Pages 35—43*

Djamel Berkane

Full paper (PDF, 164 Kb) | Abstract

We show that

where

*λ*_{d} is a real valued arithmetic function called the Barban and Vehov weight and we give an explicit version of a Theorem of Barban and Vehov which has applications to zero-density theorems.

**Mean values of the error term with shifted arguments in the circle problem**

*Original research paper. Pages 44—51*

Jun Furuya and Yoshio Tanigawa

Full paper (PDF, 196 Kb) | Abstract

In this paper, we show the relation between the shifted sum of a number-theoretic error term and its continuous mean (integral). We shall obtain a certain expression of the shifted sum as a linear combination of the continuous mean with the Bernoulli polynomials as their coefficients. As an application of our theorem, we give better approximations of the continuous mean by a shifted sum.

**On certain inequalities for ***σ*, *φ*, *ψ* and related functions

*Original research paper. Pages 52—60*

József Sándor

Full paper (PDF, 162 Kb) | Abstract

Some new inequalities for the arithmetic functions of the title are considered. Among others we offer a refinement of a recent arithmetic inequality by K. T. Atanassov {1}.

**On rational fractions not expressible as a sum of three unit fractions**

*Original research paper. Pages 61—64*

Simon Brown

Full paper (PDF, 80 Kb) | Abstract

Of those fractions (*a*/*b* < 1) that can not be expressed as a sum of three unit fractions, many can be written in terms of three unit fractions if the smallest denominator is ⎣*b* / *a*⎦ and the next largest denominator is < 0. General expressions are given for some specific classes of these. Two examples of Yamamoto are reconsidered.

**A note on a broken Dirichlet convolution**

*Original research paper. Pages 65—73*

Emil Daniel Schwab and Barnabás Bede

Full paper (PDF, 185 Kb) | Abstract

The paper deals with a broken Dirichlet convolution ⊗ which is based on using the odd divisors of integers. In addition to presenting characterizations of ⊗-multiplicative functions we also show an analogue of Menon’s identity:

where (

*a*,

*n*)

_{⊗} denotes the greatest common odd divisor of

*a* and

*n*, φ

_{⊗}(

*n*) is the number of integers

*a* (mod

*n*) such that (

*a*,

*n*)

_{⊗} = 1,

*τ*(

*n*) is the number of divisors of

*n*, and

*τ*_{2}(

*n*) is the number of even divisors of

*n*.

**On a recurrence related to 321–avoiding permutations**

*Original research paper. Pages 74—78*

Toufik Mansour and Mark Shattuck

Full paper (PDF, 166 Kb) | Abstract

Dokos et al. recently conjectured that the distribution polynomial

*f*_{n}(

*q*) on the set of permutations of size

*n* avoiding the pattern 321 for the number of inversions is given by:

with

*f*_{0}(

*q*) = 1, which was later proven in the affirmative, see {1}. In this note, we provide a new proof of this conjecture, based on the scanning-elements algorithm described in {3}, and present an identity obtained by equating two explicit formulas for the generating function

.

**Nesterenko-like rational function, useful to prove the Apéry’s theorem**

*Original research paper. Pages 79—91*

Anier Soria Lorente

Full paper (PDF, 209 Kb) | Abstract

In this paper, a brief introduction to the Apéry’s result and to the so called phenomenon of Apéry’s is given. Here, a modification of the Nesterenko’s rational function, from which new Diophantine approximations to ζ(3) are deduced, is presented. Moreover, as a consequence we deduce the corresponding Apéry-Like recurrence relation as well as a new continued fraction expansion and a new series expansion for ζ(3).

**Some arithmetic properties of an analogue of Möbius function**

*Original research paper. Pages 92—96*

Ramesh Kumar Muthumalai

Full paper (PDF, 141 Kb) | Abstract

Some properties and applications of an analogue of Möbius function are studied in the paper titled, “Some properties and application of a new arithmetic function in analytic number theory”. In this paper, some additional properties of this new arithmetic function connecting with familiar arithmetic functions such as Möbius function, Euler totient function, etc., are given.

**Volume 20** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4 ▷ Number 5

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