**Volume 28** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4 (Online First)

**On wide-output sieves**

*Original research paper. Pages 147—158*

Alessandro Cotronei

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*ω*and

*Ω*and the divisor function

*d*) to each natural number in the considered range of integer numbers. We prove that in some cases the algorithms presented have a relatively small computational complexity. A more detailed output is indeed obtained with respect to the original Sieve of Eratosthenes.

**Remark on the transcendence of real numbers generated by Thue–Morse along squares**

*Original research paper. Pages 159—166*

Eiji Miyanohara

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**A generalization of multiple zeta values. Part 1: Recurrent sums**

*Original research paper. Pages 167—199*

Roudy El Haddad

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*recurrent sums*, where the reciprocals are replaced by arbitrary sequences. We introduce a toolbox of formulas for the manipulation of such sums. We begin by developing variation formulas that allow the variation of a recurrent sum of order

*m*to be expressed in terms of lower order recurrent sums. We then proceed to derive theorems (which we will call inversion formulas) which show how to interchange the order of summation in a multitude of ways. Later, we introduce a set of new partition identities in order to then prove a reduction theorem which permits the expression of a recurrent sum in terms of a combination of non-recurrent sums. Finally, we use these theorems to derive new results for multiple zeta star values and recurrent sums of powers.

**A generalization of multiple zeta values. Part 2: Multiple sums**

*Original research paper. Pages 200—233*

Roudy El Haddad

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*multiple sums*, where the reciprocals are replaced with arbitrary sequences. We develop formulae to help with manipulating such sums. We develop variation formulae that express the variation of multiple sums in terms of lower order multiple sums. Additionally, we derive a set of partition identities that we use to prove a reduction theorem that expresses multiple sums as a combination of simple sums. We present a variety of applications including applications concerning polynomials and MZVs such as generating functions and expressions for and . Finally, we establish the connection between multiple sums and a type of sums called recurrent sums. By exploiting this connection, we provide additional partition identities for odd and even partitions.

**Note on some sequences having periods that divide ( p^{ p} − 1) / (p − 1)**

*Original research paper. Pages 234—239*

Abdelkader Benyattou and Miloud Mihoubi

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**On the equation f(n^{2} − Dnm + m^{2}) = f^{ 2}(n) − Df(n)f(m) + f^{ 2}(m)**

*Original research paper. Pages 240—251*

B. M. Phong and R. B. Szeidl

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where .

**Some equalities and binomial sums about the generalized Fibonacci number u_{n}**

*Original research paper. Pages 252—260*

Yücel Türker Ulutaş and Derya Toy

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*q*-Fibonacci bicomplex and *q*-Lucas bicomplex numbers

*Original research paper. Pages 261—275*

Fügen Torunbalcı Aydın

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*q*-Fibonacci bicomplex numbers and the

*q*-Lucas bicomplex numbers, respectively. Then, we give some algebraic properties of the

*q*-Fibonacci bicomplex numbers and the

*q*-Lucas bicomplex numbers.

**A note on the Aiello–Subbarao conjecture on addition chains**

*Original research paper. Pages 276—280*

Hatem M. Bahig

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*for each positive integer*, where denotes the minimal length of an addition chain for In 1993, Aiello and Subbarao stated the apparently stronger conjecture that

*there is an addition chain for*

*with length equals to*We note that the Aiello–Subbarao conjecture is not stronger than the Scholz (also called the Scholz–Brauer) conjecture.

**On certain rational perfect numbers**

*Original research paper. Pages 281—285*

József Sándor

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**Perfect squares in the sum and difference of balancing-like numbers**

*Original research paper. Pages 286—301*

M. K. Sahukar, Zafer Şiar, Refik Keskin and G. K. Panda

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**About the theorem that partially solves the Navarrete–Orellana Conjecture**

*Original research paper. Pages 302—317*

Jorge Andrés Julca Avila and Gabriel Silva de Andrade

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*“given a large prime number a sequence is generated, in such a way that all odd prime numbers, except the given prime, are fixed points of that sequence”*. In this work, we formulated a theorem that partially confirms the veracity of this conjecture, more specifically, all prime numbers of a given line segment are fixed points of this sequence.

**On hyperbolic k-Jacobsthal and k-Jacobsthal–Lucas octonions**

*Original research paper. Pages 319—330*

Engin Özkan and Mine Uysal

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*k*-Jacobsthal and

*k*-Jacobsthal–Lucas octonions. We give Binet’s Formula, Cassini’s identity, Catalan’s identity, d’Ocagne identity, generating functions of the hyperbolic

*k*-Jacobsthal and

*k*-Jacobsthal–Lucas octonions. Also, we present many properties of these octonions.

**Two 2-Fibonacci sequences generated by a mixed scheme. Part 1**

*Original research paper. Pages 331—338*

Krassimir T. Atanassov

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*n*-th members are given. For difference of all previous sequences from Fibonacci type, the present 2-Fibonacci sequences are obtained by a new way. It is proved that the new sequences have bases with 48 elements about function 𝜑 and modulo 9.

**The Hadamard-type k-step Pell sequences**

*Original research paper. Pages 339—349*

Ömür Deveci, Yeşim Aküzüm and Muhammad Eshaq Rashedi

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*k*-step Pell sequence by using the Hadamard-type product of characteristic polynomials of the Pell sequence and the

*k*-step Pell sequence. Also, we derive the generating matrices for these sequences, and then we obtain relationships between the Hadamard-type

*k*-step Pell sequences and these generating matrices. Furthermore, we produce the Binet formula for the Hadamard-type

*k*-step Pell numbers for the case that

*k*is odd integers and

*k*≥ 3. Finally, we derive some properties of the Hadamard-type

*k*-step Pell sequences such as the combinatorial representation, the generating function, and the exponential representation by using its generating matrix.

**Notes on efficient computation of Ramanujan cubic equations**

*Original research paper. Pages 350—375*

Peter J.-S. Shiue, Anthony G. Shannon, Shen C. Huang, Jorge E. Reyes

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**On certain equations and inequalities involving the arithmetical functions φ(n) and d(n)**

*Original research paper. Pages 376—379*

József Sándor and Saunak Bhattacharjee

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**Corrigendum to “On upper Hermite–Hadamard inequalities for geometric-convex and log-convex functions” [Notes on Number Theory and Discrete Mathematics, 2014, Vol. 20, No. 5, 25–30]**

*Corrigendum. Pages 380—381*

József Sándor

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*This volume 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-NP3/43/2021.*

**Volume 28** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4 (Online First)