# Volume 25, 2019, Number 2

Volume 25Number 1 ▷ Number 2 ▷ Number 3Number 4

Generalized arithmetic subderivative
Original research paper. Pages 1—7
Pentti Haukkanen
Full paper (PDF, 150 Kb) | Abstract

Let 0 ≠ S ⊆ ℙ. The arithmetic subderivative of n with respect to S is defined as

DS(n) = npS νp(n)/p,

where n = Πp ∈ ℙ pνp(n) ∈ ℤ+. In particular, D(n) = D(n) is the arithmetic derivative of n, and D{p}(n) = Dp(n) is the arithmetic partial derivative of n with respect to p ∈ ℙ.

For each pS, let fp be an arithmetic function. We define generalized arithmetic subderivative of n with respect to S as

DSf(n) = npS fp(n)/p,

where f stands for the collection (fp)pS of arithmetic functions. In this paper, we examine for which kind of functions fp the generalized arithmetic subderivative is obeys the Leibniz-rule, preserves addition, “usual multiplication” and “scalar multiplication”.

Straddled numbers: numbers equal to the sum of powers of consecutive primes from the least prime factor to the largest prime factor
Original research paper. Pages 8—15
Miroslav Kureš
Full paper (PDF, 231 Kb) | Abstract

Positive integers equal to the sum of powers of consecutive primes from the least prime factor to the largest prime factor are studied. They are called the straddled numbers and their properties are derived. There are also presented some findings of such numbers and asymptotic expansions are used, too.

On the prime factors of a quasiperfect number
Original research paper. Pages 16—21
V. Siva Rama Prasad and C. Sunitha
Full paper (PDF, 174 Kb) | Abstract

A positive integer N is said to be quasiperfect if σ(N) = 2N + 1 where σ(N) is the sum of the positive divisors of N. So far no quasiperfect number is known. If such N exists, let γ(N) denote the product of the distinct primes dividing N. In this paper, we obtain a lower bound for γ(N) in terms of r = ω(N), the number of distinct prime factors of N. Also we show that every quasiperfect number N is divisible by a prime p with (i) p ≡ 1 (mod 4); (ii) p ≡ 1 (mod 5) if 5 ∤ N and (iii) p ≡ 1 (mod 3), if 3 ∤ N.

Eisenstein’s criterion, Fermat’s last theorem, and a conjecture on powerful numbers
Original research paper. Pages 22—29
Pietro Paparella
Full paper (PDF, 4121 Kb) | Abstract

Given integers > m > 0, monic polynomials Xn, Yn, and Zn are given with the property that the complex number ρ is a zero of Xn if and only if the triple (ρ, ρ + m, ρ + ) satisfies xn + yn = zn. It is shown that the irreducibility of these polynomials implies Fermat’s last theorem. It is also demonstrated, in a precise asymptotic sense, that for a majority of cases, these polynomials are irreducible via application of Eisenstein’s criterion. We conclude by offering a conjecture on powerful numbers.

Inequalities between the arithmetic functions φ, ψ and σ. Part 2
Original research paper. Pages 30—35
József Sándor and Krassimir Atanassov
Full paper (PDF, 161 Kb) | Abstract

We prove inequalities related to φ(n)φ(n).ψ(n)ψ(n) or φ(n)ψ(n).ψ(n)φ(n) and related powers, where φ and ψ denote the Euler, resp. Dedekind arithmetic functions. More general theorem for the arithmetical functions f, g and h is formulated and proved.

Remark on sk, t-Jacobsthal numbers
Original research paper. Pages 36—39
Apisit Pakapongpun
Full paper (PDF, 130 Kb) | Abstract

A new generalization of the Jacobsthal numbers is introduced and the properties of the new numbers are studied.

Indispensable digits for digit sums
Original research paper. Pages 40—48
Ji Young Choi
Full paper (PDF, 181 Kb) | Abstract

Let b be an integer greater than 1 and g = b − 1. For any nonnegative integer n, we define indispensable digits in the base-b representation of n so that we can calculate the digit sum of the base-b representation of g · n: Instead of adding every digit in it, we multiply g by the number of the indispensable digits in the base-b representation of n. Then, we find the formula to calculate the digit sum of g · n + r using the number of indispensable digits in n, for any nonnegative integers n and r with 0 < r < g.

Diophantine equations related to reciprocals of linear recurrence sequences
Original research paper. Pages 49—56
H. R. Hashim and Sz. Tengely
Full paper (PDF, 195 Kb) | Abstract

k=0 Uk(P1, Q1) / xk+1 = ∑k=0 Uk(P2, Q2) / yk+1,

in integers (x, y), where Un(P, Q) is a Lucas sequence defined by U0 = 0, U1 = 1, Un = PUn−1QUn−2 for n > 1. We also deal with a similar equation related to the generalized Tribonacci sequence.

The Gelin–Cesàro identity in some third-order Jacobsthal sequences
Original research paper. Pages 57—67
Gamaliel Cerda-Morales
Full paper (PDF, 173 Kb) | Abstract

In this paper, we deal with two families of third-order Jacobsthal sequences. The first family consists of generalizations of the Jacobsthal sequence. We show that the Gelin–Cesàro identity is satisfied. Also, we define a family of generalized third-order Jacobsthal sequences {𝕁n(3)}n ≥ 0 by the recurrence relation

𝕁n+3(3) = 𝕁n+2(3) + 𝕁n+1(3) + 2𝕁n(3), n ≥ 0,

with initials conditions 𝕁0(3) = a, 𝕁1(3) = b and 𝕁2(3) = c, where a, b and c are non-zero real numbers. Many sequences in the literature are special cases of this sequence. We find the generating function and Binet’s formula of the sequence. Then we show that the Cassini and Gelin–Cesàro identities are satisfied by the indices of this generalized sequence.

The 3 × 3 × … × 3 Points Problem solution
Original research paper. Pages 68—75
Marco Ripà
Full paper (PDF, 1501 Kb) | Abstract

We offer the first proof of Ripà’s 3 × 3 × 3 × 3 Dots Problem, providing a general solution of the 3k case (3k points arranged in a 3 × 3 × … × 3 grid), for any k ∈ ℕ − {0}. We give also new bounds for the n × n × n problem, improving many of the previous results.

A note on partially degenerate Legendre–Genocchi polynomials
Original research paper. Pages 76—90
N. U. Khan, T. Kim and T. Usman
Full paper (PDF, 195 Kb) | Abstract

In the past years, many researchers have worked on degenerate polynomials and a variety of its extentions and variants can be found in literature. Following up, in this article, we firstly introduce the partially degenerate Legendre–Genocchi polynomials, and further define a new generalization of degenerate Legendre–Genocchi polynomials. From our generalization, we establish some implicit summation formulae and symmetry identities by the generating function of partially degenerate Legendre–Genocchi polynomials. Eventually, it can be found that some recently demonstrated summations and identities stated in the article, are special cases of our results.

Original research paper. Pages 91—112
S. G. Rayaguru, D. Savin and G. K. Panda
Full paper (PDF, 249 Kb) | Abstract

Fibonacci and Lucas symbol elements are generalized to Horadam symbol elements and some properties are studied. In the last section we use these properties for to find zero divisors in symbol algebras over cyclotomic fields of finite fields.

Applications of Mollie Horadam’s generalized integers to Fermatian and Fibonacci numbers
Original research paper. Pages 113—126
A. G. Shannon
Full paper (PDF, 153 Kb) | Abstract

This paper extends some of the arithmetic functions which Mollie Horadam developed for sequences of generalized integers and apply them to some particular integer sequences, particularly the Fibonacci and Fermatian numbers.

Generalised Beatty sets
Original research paper. Pages 127—135
Marc Technau
Full paper (PDF, 196 Kb) | Abstract

Generalised Beatty sets, that is, sets of the form {⌊1 + 2 + β⌋ : m, n ∈ ℕ}, are studied, where ⌊ξ⌋ denotes the largest integer less than or equal to ξ. Such sets are shown to be contained in a suitable ordinary Beatty set {⌊ + β⌋ : n ∈ ℕ} and equal said set save for finitely many exceptions. Moreover, bounds for the largest such exception are given.

Real-rooted polynomials via generalized Bell umbra
Original research paper. Pages 136—144
Full paper (PDF, 196 Kb) | Abstract

In this paper, by the generalized Bell umbra and Rolle’s theorem, we give some results on the real rootedness of polynomials. Some applications on partition polynomials are considered. Our results are illustrated by some comprehensive examples.

Some combinatorial identities for the r-Dowling polynomials
Original research paper. Pages 145—154
Mark Shattuck
Full paper (PDF, 201 Kb) | Abstract

Recently, three new Bell number formulas were proven using algebraic methods, one of which extended an earlier identity of Gould–Quaintance and another a previous identity of Spivey. Here, making use of combinatorial arguments to establish our results, we find generalizations of these formulas in terms of the r-Dowling polynomials. In two cases, weights of the form ai and bj may be replaced by arbitrary sequences of variables xi and yj which yields further generalizations. Finally, a second extension of one of the formulas is found that involves generalized Stirling polynomials and leads to analogues of this formula for other counting sequences.

On addition-subtraction chains of numbers with low Hamming weight
Original research paper. Pages 155—168
Full paper (PDF, 200 Kb) | Abstract

An addition chain is a sequence of integers such that every element in the sequence is the sum of two previous elements. They have been much studied, and generalized to addition-subtraction chains, Lucas chains, and Lucas addition-subtraction chains. These various chains have been useful in finding efficient exponentiation algorithms in groups. As a consequence, finding chains of minimal length is critical. The main objective of this paper is to extend results known for addition chains to addition-subtraction chains with Lucas addition-subraction as a tool to construct such minimal chains. Specifically, if we let 𝓁(n) stand for the minimal length of all the Lucas addition-subtraction chains for n, we prove |𝓁(2n) − 𝓁(n)| ≤ 1 for all integers n of Hamming weight ≤ 4. Thus, to find a minimal addition-subtraction chains for low Hamming weight integers, it suffices to only consider odd integers.

Identities for generalized balancing numbers
Original research paper. Pages 169—180
Robert Frontczak
Full paper (PDF, 190 Kb) | Abstract

We study properties of generalized balancing numbers. We start with some basic identities. Thereafter, we focus on connections to generalized Fibonacci numbers. Using generating functions we prove fundamental relations between these two sequences. Many interesting examples involving balancing, Lucas-balancing, Fibonacci, and Lucas numbers are obtained as special cases of our relations.

Interior vertices and edges in bargraphs
Original research paper. Pages 181—189
Toufik Mansour and Armend Sh. Shabani
Full paper (PDF, 198 Kb) | Abstract

In this paper, we consider two statistics on bargraphs, which are defined to be lattice paths in the first quadrant, starting at the origin and ending upon first return to the x-axis. Each bargraph is represented as a sequence of columns π1π2πm such that column k contains πk cells. First we enumerate interior vertices, where naturally, interior vertex is a vertex that belongs to exactly four cells of bargraphs. Then we enumerate d-edges – edges that contain d interior vertices. More precisely, we find the generating function for the number of bargraphs with n cells and m columns according: to interior vertices and according to horizontal (vertical) d-edges. In addition we consider several special cases in detail, where we obtain asymptotic results for total number of statistics under consideration.

Equitable coloring on subdivision vertex join of cycle Cm with path Pn
Original research paper. Pages 190—198
K. Praveena, M. Venkatachalam and A. Rohini
Full paper (PDF, 155 Kb) | Abstract

Graph coloring is one of the research areas that shaped the graph theory as we know it today. An equitable coloring of a graph G is a proper coloring of the vertices of G such that color classes differ in size by at most one. The subdivision graph S(G) of a graph G is the graph obtained by inserting a new vertex into every edge of G. Let G1 and G2 be two graphs with vertex sets V(G1) and V(G2), respectively. The subdivision-vertex join of two vertex disjoint graphs G1 and G2 is the graph obtained from S(G1) and G2 by joining each vertex of V(G1) with every vertex of V(G2). In this paper, we find the equitable chromatic number of subdivision vertex join of cycle graph with path graph.

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. KP-06-NP-28/2018.

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