**Volume 25** ▶ Number 1 ▷ Number 2

**Generalized arithmetic subderivative**

*Original research paper. Pages 1—7*

Pentti Haukkanen

Full paper (PDF, 150 Kb) | Abstract

*S*⊆ ℙ. The arithmetic subderivative of

*n*with respect to

*S*is defined as

*D*_{S}(*n*) = *n* ∑_{p∈S} *ν*_{p}(*n*)/*p*,

where *n* = Π_{p ∈ ℙ} *p*^{νp(n)} ∈ ℤ_{+}. In particular, *D*_{ℙ}(*n*) = *D*(*n*) is the arithmetic derivative of *n*, and *D*_{{p}}(*n*) = *D*_{p}(*n*) is the arithmetic partial derivative of *n* with respect to *p* ∈ ℙ.

For each *p* ∈ *S*, let *f _{p}* be an arithmetic function. We define generalized arithmetic subderivative of

*n*with respect to

*S*as

*D _{S}^{f}*(

*n*) =

*n*∑

_{p∈S}

*f*(

_{p}*n*)/

*p*,

where *f* stands for the collection (*f _{p}*)

_{p∈S}of arithmetic functions. In this paper, we examine for which kind of functions

*f*the generalized arithmetic subderivative is obeys the Leibniz-rule, preserves addition, “usual multiplication” and “scalar multiplication”.

_{p}**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

**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

*N*is said to be

*quasiperfect*if

*σ*(

*N*) = 2

*N*+ 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

*ℓ*>

*m*> 0, monic polynomials

*X*,

_{n}*Y*, and

_{n}*Z*are given with the property that the complex number

_{n}*ρ*is a zero of

*X*if and only if the triple (

_{n}*ρ*,

*ρ*+

*m*,

*ρ*+

*ℓ*) satisfies

*x*+

^{n}*y*=

^{n}*z*. 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.

^{n}**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

*φ*(

*n*)

^{φ(n)}.

*ψ*(

*n*)

^{ψ(n)}or

*φ*(

*n*)

^{ψ(n)}.

*φ*(

*n*)

^{ψ}

^{(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 s_{k}, t-Jacobsthal numbers**

*Original research paper. Pages 36—39*

Apisit Pakapongpun

Full paper (PDF, 130 Kb) | Abstract

**Indispensable digits for digit sums**

*Original research paper. Pages 40—48*

Ji Young Choi

Full paper (PDF, 181 Kb) | Abstract

*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}^{∞} *U _{k}*(

*P*

_{1},

*Q*

_{1}) /

*x*

^{k+1}= ∑

_{k=0}

^{∞}

*U*(

_{k}*P*

_{2},

*Q*

_{2}) /

*y*

^{k+1},

in integers (*x*, *y*), where *U _{n}*(

*P, Q*) is a Lucas sequence defined by

*U*

_{0}= 0,

*U*

_{1}= 1,

*U*=

_{n}*PU*

_{n−1}−

*QU*

_{n−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

_{n}

^{(3)}}

_{n ≥ 0}by the recurrence relation

𝕁_{n+3}^{(3)} = 𝕁_{n+2}^{(3)} + 𝕁_{n+1}^{(3)} + 2𝕁_{n}^{(3)}, *n* ≥ 0,

with initials conditions 𝕁_{0}^{(3)} = *a*, 𝕁_{1}^{(3)} = *b* and 𝕁_{2}^{(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

^{k}case (3

^{k}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

**On some Horadam symbol elements**

*Original research paper. Pages 91—112*

S. G. Rayaguru, D. Savin and G. K. Panda

Full paper (PDF, 249 Kb) | Abstract

**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

**Generalised Beatty sets**

*Original research paper. Pages 127—135*

Marc Technau

Full paper (PDF, 196 Kb) | Abstract

*mα*

_{1}+

*nα*

_{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 {⌊

*nα*+

*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*

Abdelkader Benyattou and Miloud Mihoubi

Full paper (PDF, 196 Kb) | Abstract

**Some combinatorial identities for the r-Dowling polynomials**

*Original research paper. Pages 145—154*

Mark Shattuck

Full paper (PDF, 201 Kb) | Abstract

*r*-Dowling polynomials. In two cases, weights of the form

*a*and

^{i}*b*may be replaced by arbitrary sequences of variables

^{j}*x*and

_{i}*y*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.

_{j}**On addition-subtraction chains of numbers with low Hamming weight**

*Original research paper. Pages 155—168*

Dustin Moody and Amadou Tall

Full paper (PDF, 200 Kb) | Abstract

^{−}(

*n*) stand for the minimal length of all the Lucas addition-subtraction chains for

*n*, we prove |𝓁

^{−}(2

*n*) − 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

**Interior vertices and edges in bargraphs**

*Original research paper. Pages 181—189*

Toufik Mansour and Armend Sh. Shabani

Full paper (PDF, 198 Kb) | Abstract

*x*-axis. Each bargraph is represented as a sequence of columns

*π*

_{1}

*π*

_{2}…

*π*

_{m}such that column

*k*contains

*π*

_{k}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 C_{m} with path P_{n}**

*Original research paper. Pages 190—198*

K. Praveena, M. Venkatachalam and A. Rohini

Full paper (PDF, 155 Kb) | Abstract

*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

*G*

_{1}and

*G*

_{2}be two graphs with vertex sets

*V*(

*G*

_{1}) and

*V*(

*G*

_{2}), respectively. The subdivision-vertex join of two vertex disjoint graphs

*G*

_{1}and

*G*

_{2}is the graph obtained from

*S*(

*G*

_{1}) and

*G*

_{2}by joining each vertex of

*V*(

*G*

_{1}) with every vertex of

*V*(

*G*

_{2}). 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.*