**Volume 24** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Some formulae which match with the prime counting function infinitely often **

*Original research paper. Pages 1–4*

Yannick Saouter

Full paper (PDF, 157 Kb) | Abstract

**On Dris conjecture about odd perfect numbers **

*Original research paper. Pages 5–9*

Paolo Starni

Full paper (PDF, 141 Kb) | Abstract

*n*=

*π*

^{α}N^{2}, where

*π*is prime, (

*π*,

*N*) = 1 and

*π*≡

*α*≡ 1 (mod 4). Dris conjecture states that

*N*>

*π*. We find that

^{α}*N*

^{2}> 1/2

*π*, with

^{γ}*γ*= max{

*ω*(

*n*) − 1,

*α*};

*ω*(

*n*) ≥ 10 is the number of distinct prime factors of

*n*.

**An indicator characteristic for twin prime formation independent of integer size **

*Original research paper. Pages 10–15*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 199 Kb) | Abstract

*Z*

_{6}has twin primes located in the same row. This enables the structural mechanisms underlying the formation of twin primes to be summarised by simple equations. The classification system provided by right-end-digits applies equally in all integer domains of any size, and can be used to demonstrate the formation of twin primes in such domains.

**On a generalization of Eulerian numbers **

*Original research paper. Pages 16–42*

Claudio Pita-Ruiz

Full paper (PDF, 261 Kb) | Abstract

*rp*-th degree

*n*-polynomial , where

*a*,

*b*∈ ℂ,

*a*≠ 0,

*r*,

*p*∈ ℕ, and the -th degree n-polynomial , where

*α*,

_{s}*β*∈ ℂ,

_{s}*r*,

_{s}*p*∈ ℕ,

_{s}*s*= 2, …,

*l*. In the expansion of the polynomial in terms of the binomials , , the resulting coefficients are the generalized Eulerian numbers we consider in this work (the case

*P*(

*n*) = 1,

*a*= 1,

*b*= 0

*, r*= 1 corresponds to the standard Eulerian numbers). We obtain results on symmetries, recurrences, row sums, and alternating row sums, that generalize the corresponding well-known results for the standard Eulerian numbers. The main tool we use to obtain our results throughout the work, is the Z-transform of sequences.

**Proofs of certain conjectures for means of two arguments **

*Original research paper. Pages 43–48*

József Sándor

Full paper (PDF, 150 Kb) | Abstract

**A new proof of Euler’s pentagonal number theorem **

*Original research paper. Pages 49–52*

A. David Christopher

Full paper (PDF, 163 Kb) | Abstract

**New index matrix representations of operations over natural numbers **

*Original research paper. Pages 53–60*

Lilija Atanassova

Full paper (PDF, 166 Kb) | Abstract

**Almost arithmetic progressions in ℤ/pℤ **

*Original research paper. Pages 61–75*

Mario Huicochea

Full paper (PDF, 202 Kb) | Abstract

*k*∈ ℕ ∪ {0} and

*r*∈ ℤ/

*p*ℤ \ {0}, we say that a subset

*X*of ℤ/

*p*ℤ is a

*k*-almost arithmetic progression with difference r if there is an arithmetic progression

*Y*with difference r containing

*X*such that |

*Y*\

*X*| ≤

*k*. Let

*X*be a

*k*-almost arithmetic progression with difference r such that

*k*+ 2 < |

*X*| <

*p*− 4

*k*− 9. The main result of this paper is following: if there is

*t*∈ ℤ/

*p*ℤ \ {0} such that

*X*is also a

*k*-almost arithmetic progression with difference

*t*, then

*t*∈ {±

*r*}. Moreover, we will show that our result is sharp.

**Fourier series of sums of products of Bernoulli and Euler/Genocchi functions **

*Original research paper. Pages 76–93*

Taekyun Kim, Dae San Kim, Toufik Mansour and Gwan-Woo Jang

Full paper (PDF, 282 Kb) | Abstract

**Short note on a new arithmetic function **

*Original research paper. Pages 94–96*

Krassimir T. Atanassov

Full paper (PDF, 136 Kb) | Abstract

*π*– and

*φ*-functions.

**The greatest common divisors of generalized Fibonacci and generalized Pell numbers **

*Original research paper. Pages 97–102*

Boonyen Thongkam and Nutcha Sailadda

Full paper (PDF, 141 Kb) | Abstract

*U*)

_{n}^{a,b,r}_{n≥0}such that sequence generalizes both generalized Fibonacci numbers (

*G*)

_{n}^{a,b}_{n≥0}and generalized Pell numbers (

*P*)

_{n}^{a,b}_{n≥0}. In the present paper, we show a study of the greatest common divisors of some

*G*,

_{n}^{a,b}*P*and

_{n}^{a,b}*U*.

_{n}^{a,b,r}**Fibonacci and Lucas numbers via the determinants of tridiagonal matrix **

*Original research paper. Pages 103–108*

Taras Goy

Full paper (PDF, 167 Kb) | Abstract

**The arrowhead-Pell-random-type sequences**

*Original research paper. Pages 109–119*

Özgür Erdağ, Anthony G. Shannon and Ömür Deveci

Full paper (PDF, 210 Kb) | Abstract

**On generalizations of the Jacobsthal sequence **

*Original research paper. Pages 120–135*

Fügen Torunbalcı Aydın

Full paper (PDF, 227 Kb) | Abstract

**Corrigendum to “Sum of dilates of two sets” [Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 4, 34–41] **

*Corrigendum. Page 136*

Raj Kumar Mistri

Corrigendum (PDF, 47 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. DNP-06-38/2017.*