**Volume 17** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**The relation between π(x) and certain arithmetic functions**

*Original research paper. Pages 1–9*

Magdalena Corciovei-Bănescu

Full paper (PDF, 178 Kb) | Abstract

where

*A*(

_{k}*x*) = Σ

*and*

_{p ≤ x}p^{k}*k*≥ 0.

** A characterization of modularity in graphs**

*Original research paper. Pages 10–12*

Yilun Shang

Full paper (PDF, 121 Kb) | Abstract

** Note on φ, ψ and σ-functions. Part 3**

*Original research paper. Pages 13–14*

Krassimir Atanassov

Full paper (PDF, 96 Kb) | Abstract

*φ*,

*ψ*and

*σ*–functions are formulated and proved.

** On a class of infinite sequences with relatively prime numbers and twin prime conjecture**

*Original research paper. Pages 15–17*

Blagoy N. Djokov

Full paper (PDF, 116 Kb) | Abstract

**On the Diophantine equation y^{n} = f(x)^{n} + g(x)**

*Original research paper. Pages 18–21*

R. Srikanth and S. Subburam

Full paper (PDF, 131 Kb) | Abstract

** A note on switching in symmetric n–sigraphs**

*Original research paper. Pages 22–25*

P. Siva Kota Reddy, B. Prashanth and Kavita S. Permi

Full paper (PDF, 138 Kb) | Abstract

*n*-sigraphs.

**The structure of even powers in Z_{3}: Critical structural factors that prevent the formation of even–powered triples greater than squares**

*Original research paper. Pages 26–30*

J. V. Leyendekkers and A. Shannon

Full paper (PDF, 119 Kb) | Abstract

*N*

^{4m}+

*M*

^{4m}) ever to equal an equivalent power. The number 3 plays a vital role as integers divisible by 3, when raised to an even power of the form 4

*m*, have rows in a table of modular rings which are triangular numbers, whereas other integers raised to the same power have rows which are pentagonal numbers. The substructure within these sequences of pentagonal numbers is order within order, analogous to structure in chaos theory.

** The structure of geometric number sequences**

*Original research paper. Pages 31–37*

J. V. Leyendekkers and A. Shannon

Full paper (PDF, 50 Kb) | Abstract

_{3}and Z

_{4}of some elements of these sequences reinforce previous studies of their properties.

** Some properties and applications of a new arithmetic function in analytic number theory**

*Original research paper. Pages 38–48*

Ramesh Kumar Muthumalai

Full paper (PDF, 189 Kb) | Abstract

** Some properties of unitary addition Cayley graphs**

*Original research paper. Pages 49–59*

Deepa Sinha, Pravin Garg and Anjali Singh

Full paper (PDF, 180 Kb) | Abstract

*B*be a subset of Γ. The addition Cayley graph

*G*′ = Cay

^{+}(Γ,

*B*) is the graph having the vertex set

*V*(

*G*′) = Γ and the edge set

*E*(

*G*′) = {

*ab*:

*a*+

*b*∈

*B*}, where

*a*,

*b*∈ Γ. For a positive integer

*n*> 1, the unitary addition Cayley graph

*G*is the graph whose vertex set is

_{n}*Z*, the integers modulo

_{n}*n*and if

*U*denotes set of all units of the ring

_{n}*Z*, then two vertices a, b are adjacent if and only if

_{n}*a + b ∈ U*. The unitary addition Cayley graph

_{n}*G*is also defined as,

_{n}*G*= Cay

_{n}^{+}(

*Z*,

_{n}*U*). In this paper, we discuss the several properties of unitary addition Cayley graphs and also obtain the characterization of planarity and outerplanarity of unitary addition Cayley graphs.

_{n}** Erratum to “Modular rings and the integer 3”**

*Erratum. Page 60*

Erratum (PDF, 10 Kb)