**Volume 28** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Bounds on some energy-like invariants of corona and edge corona of graphs**

*Original research paper. Pages 383—398*

Chinglensana Phanjoubam and Sainkupar Mn Mawiong

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**On linear algebra of one type of symmetric matrices with harmonic Fibonacci entries**

*Original research paper. Pages 399—410*

Mücahit Akbıyık, Seda Yamaç Akbıyık and Fatih Yılmaz

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**Bi-unitary multiperfect numbers, IV(c)**

*Original research paper. Pages 411—434*

Pentti Haukkanen

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Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers. The present paper is part IV(c) in a series of papers on even bi-unitary multiperfect numbers. In parts I, II and III we determined all bi-unitary triperfect numbers of the form , where and is odd. In part V we fixed the case . The case is more difficult. In Parts IV(a-b) we solved partly this case, and in the present paper (Part IV(c)) we continue the study of the same case ().

**Note on the natural density of r-free numbers**

*Original research paper. Pages 435—440*

Sunanta Srisopha, Teerapat Srichan and Sukrawan Mavecha

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*P*be a finite set of prime numbers. By using an elementary method, the proportion of all

*r*-free numbers which are divisible by at least one element in

*P*is studied.

**Number of stable digits of any integer tetration**

*Original research paper. Pages 441—457*

Marco Ripà and Luca Onnis

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**On complex Leonardo numbers**

*Original research paper. Pages 458—465*

Adnan Karataş

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**Binomial sums with k-Jacobsthal and k-Jacobsthal–Lucas numbers**

*Original research paper. Pages 466—476*

A. D. Godase

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*k*-Jacobsthal and

*k*-Jacobsthal–Lucas numbers. Moreover, we use multinomial theorem to obtain distinct binomial sums of

*k*-Jacobsthal and

*k*-Jacobsthal–Lucas numbers.

**Generalized Pisano numbers**

*Original research paper. Pages 477—490*

Yüksel Soykan, İnci Okumuş and Erkan Taşdemir

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**On the derivatives of B-Tribonacci polynomials**

*Original research paper. Pages 491—499*

Suchita Arolkar

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**Average value of some certain types of arithmetic functions with Piatetski-Shapiro sequences**

*Original research paper. Pages 500—506*

Suphawan Janphaisaeng, Teerapat Srichan and Pinthira Tangsupphathawat

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**Some aspects of interchanging difference equation orders**

*Original research paper. Pages 507—516*

Anthony G. Shannon and Engin Özkan

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**On edge irregularity strength of line graph and line cut-vertex graph of comb graph**

*Original research paper. Pages 517—524*

H. M. Nagesh and V. R. Girish

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**On certain rational perfect numbers, II**

*Original research paper. Pages 525—532*

József Sándor

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**Some congruences on the hyper-sums of powers of integers involving Fermat quotient and Bernoulli numbers**

*Original research paper. Pages 533—541*

Fouad Bounebirat and Mourad Rahmani

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*p*≥ 5, let ℤ

_{p}denote the set of rational

*p*-integers (those rational numbers whose denominator is not divisible by

*p*). In this paper, we establish some congruences modulo a prime power

*p*

^{5}on the hyper-sums of powers of integers in terms of Fermat quotient, Wolstenholme quotient, Bernoulli and Euler numbers.

**An introduction to harmonic complex numbers and harmonic hybrid Fibonacci numbers: A unified approach**

*Original research paper. Pages 542—557*

Emel Karaca and Fatih Yılmaz

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**Objects generated by an arbitrary natural number. Part 2: Modal aspect**

*Original research paper. Pages 558—563*

Krassimir T. Atanassov

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*Set*(

*n*), generated by an arbitrary natural number

*n*, was defined in [2] and some arithmetic functions, defined over its elements are introduced in an algebraic aspect. Here, over the elements of

*Set*(

*n*), two arithmetic functions similar to the modal type of operators are defined and some of their basic properties are studied.

**Identities involving some special numbers and polynomials on p-adic integral**

*Original research paper. Pages 564—574*

Neşe Ömür, Sibel Koparal, Ömer Duran and Kübra Nur Südemen

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*p*-adic integrals and combinatorial techniques.

**On a new additive arithmetic function related to a fixed integer**

*Original research paper. Pages 575—580*

Mihoub Bouderbala and Meselem Karras

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such that denotes the greatest common divisor of the integers and .

**Eisenstein series of level 6 and level 10 with their applications to theta function identities of Ramanujan**

*Original research paper. Pages 581—588*

A. I. Vijaya Shankar

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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.*