**Volume 22** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Professor József Sándor at σ(59) years**

*Editorial. Pages 1—11 *

Krassimir T. Atanassov

Editorial (PDF, 460 Kb)

**(RETRACTED) A note on prime zeta function and Riemann zeta function**

*Original research paper. Pages 12—15*

Mladen Vassilev–Missana

RETRACTED Full paper (PDF, 135 Kb) | Retraction Notice

As of June 2021, the paper “A note on prime zeta function and Riemann zeta function” by Mladen Vassilev–Missana, published in Notes on Number Theory and Discrete Mathematics, Vol. 22, 2016, No. 4, 12–15, has been retracted.

The reasons for retraction are identified errors, as pointed out by Richard P. Brent in the paper “On two theorems of Vassilev-Missana”, submitted on 23 March 2021 and immediately published in the June issue of Notes on Number Theory and Discrete Mathematics, Vol. 27, 2021, No. 2, 49–50, DOI: 10.7546/nntdm.2021.27.2.49-50. http://nntdm.net/volume-27-2021/number-2/49-50/

Correction entitled “A note on prime zeta function and Riemann zeta function. Corrigendum” was provided by Mladen Vassilev–Missana on 21 April 2021 and published in the same issue of Notes on Number Theory and Discrete Mathematics, Vol. 27, 2021, No. 2, 51–53, DOI: 10.7546/nntdm.2021.27.2.51-53. http://nntdm.net/volume-27-2021/number-2/51-53/

The electronic copy of the retracted paper is retained on the Journal website in order to maintain the scientific record, with the additional “Retracted Paper” watermark and the accompanying Retraction Notice, which as of June 2021 must be considered an integral part of the publication.

The Publisher apologizes for any inconvenience caused!

**An arithmetic function decreasing the natural numbers**

*Original research paper. Pages 16—19 *

Krassimir T. Atanassov

Full paper (PDF, 130 Kb) | Abstract

**On certain logarithmic inequalities**

*Original research paper. Pages 20—24 *

József Sándor

Full paper (PDF, 139 Kb) | Abstract

**Balancing sequence contains no prime number**

*Original research paper. Pages 25—28 *

Shekh Mohammed Zahid

Full paper (PDF, 54 Kb) | Abstract

*n*– 1) = (

*n*+ 1) + (

*n*+ 2) + … + (

*n*+

*r*), where

*n*and

*r*are positive integers. The pairs (

*n*,

*r*) constitute a solution of above equation then

*n*is called balancing number and

*r*is the corresponding balancer. In this paper, we prove a main result that there is no prime number in the sequence of balancing numbers.

**Asymptotic formulae for the number of repeating prime sequences less than N**

*Original research paper. Pages 29—40*

Christopher L. Garvie

Full paper (PDF, 233 Kb) | Abstract

*p*,

*p*+2), the prime triplet conjecture, (

*p*,

*p*+2,

*p*+6) are simple examples, are true and that prime sequences of arbitrary length can be found and shown to repeat indefinitely. Asymptotic formulae comparable to the prime number theorem are derived for arbitrary length sequences. An elementary proof is also derived for the prime number theorem and Dirichlet’s Theorem on the arithmetic progression of primes.

**Identities for balancing numbers using generating function and some new congruence relations**

*Original research paper. Pages 41—48 *

Prasanta K Ray, Sunima Patel and Manoj K Mandal

Full paper (PDF, 173 Kb) | Abstract

*n*− 1) = (

*n*+ 1) + (

*n*+ 2) + … + (

*n*+

*r*), where

*r*is the balancer corresponding to the balancing number

*n*. Thus if

*n*is a balancing number, then 8

*n*

^{2}+ 1 is a perfect square and its positive square root is called a Lucas-balancing number. The goal of this paper is to establish some new identities of these numbers.

**Figurate numbers in the modular ring Z _{4}**

*Original research paper. Pages 49—55*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 164 Kb) | Abstract

_{4}, ̅3

_{4}⊂ Z

_{4}, a modular ring, show clear distinctions between the two classes. In particular, the sum for class ̅1

_{4}is related to the Golden Ratio family of sequences, and in this class when the position of an odd integer is a prime number, then the sum always has a factor of 6. Sums of the primes in these classes can be primes but the structures are quite different, and no sums of odd integers in general are primes. The sums are related to the sequences of triangular numbers and hexagonal numbers.

**Right circulant determinant sequences with Jacobsthal and Jacobsthal–Lucas numbers**

*Original research paper. Pages 56—61*

Aldous Cesar F. Bueno

Full paper (PDF, 161 Kb) | Abstract

*J*

_{s+t}while the other makes use of Jacobsthal–Lucas numbers of the form

*K*

_{s+t}, where

*s*,

*t*∈ ℤ and

*s*≠

*t*. We also give some open problems.

**Upper and lower bounds for π based on Vieta’s geometrical approach**

*Original research paper. Pages 62—72 *

Krishna Busawon

Full paper (PDF, 538 Kb) | Abstract

**Landau’s Fourth problem**

*Original research paper. Pages 73—77 *

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 77 Kb) | Abstract

*n*

^{2}+ 1 show no deviations contrary to the natural integer structure within the modular ring Z

_{4}and the sum of two squares. Hence primes of this form should occur to infinity with other primes. Trend characteristics of primes and composites were compared graphically.

**Infinite product involves the Tribonacci numbers**

*Original research paper. Pages 78—81 *

Kantaphon Kuhapatanakul, Pornpawee Anantakitpaisal, Chanokchon Onsri and Suriya Na nhongkai

Full paper (PDF, 127 Kb) | Abstract

_{k=n}

^{∞}(1 − 1/

*T*), where

_{k}*T*are the Tribonacci numbers. We also consider a similar formula for the Tribonacci numbers with indices in arithmetic progression and give an open problem of the Diophantine equation about the Tribonacci numbers.

_{n}**On the metric dimension of the total graph of a graph**

*Original research paper. Pages 82—95 *

B. Sooryanarayana, Shreedhar K. and Narahari N.

Full paper (PDF, 263 Kb) | Abstract

*G*is a set

*S*⊆

*V*(

*G*), such that, every pair of distinct vertices of

*G*is resolved by some vertex in S. The metric dimension of

*G*, denoted by

*β*(

*G*), is the minimum cardinality of all the resolving sets of G. Shamir Khuller et al. [10], in 1996, proved that a graph

*G*with

*β*(

*G*) = 2 can have neither

*K*

_{5}nor

*K*

_{3,3}as its subgraph. In this paper, we obtain a forbidden subgraph, other than

*K*

_{5}and

*K*

_{3,3}, for a graph with metric dimension two. Further, we obtain the metric dimension of the total graph of some graph families. We also establish a Nordhaus–Gaddum type inequality involving the metric dimensions of a graph and its total graph and obtain the metric dimension of the line graph of the two dimensional grid

*P*×

_{m}*P*.

_{n}