**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

**Fibonacci primes**

*Original research paper. Pages 6—9*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 85 Kb) | Abstract

*F*, have factors (

_{p}*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*and a function of

_{p}*F*provide an approximate check on primality.

_{p}**Fibonacci primes of special forms**

*Original research paper. Pages 10—19*

Diana Savin

Full paper (PDF, 202 Kb) | Abstract

*x*

^{2}+

*ry*

^{2}(where

*r*= 1;

*r*= prime or

*r*= perfect power) is provided.

**(RETRACTED) A Diophantine system about equal sum of cubes**

*Pages 20—28*

Zhi Ren

**Retraction Notice**

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*.

The Publisher apologizes for any inconvenience caused!

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

*Original research paper. Pages 29—34*

Susil Kumar Jena

Full paper (PDF, 138 Kb) | Abstract

*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

where

*λ*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.

_{d}**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

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

*Original research paper. Pages 52—60*

József Sándor

Full paper (PDF, 162 Kb) | Abstract

**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

*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

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

*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

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

*Original research paper. Pages 92—96*

Ramesh Kumar Muthumalai

Full paper (PDF, 141 Kb) | Abstract

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

Pingback: Editorial notice, April 2016 | Notes on Number Theory and Discrete Mathematics