**Volume 26** ▶ Number 1 ▷ Number 2 ▷ Number 3

**On the equation φ(n) + d(n) = n and related inequalities**

*Original research paper. Pages 1—4*

József Sándor

Full paper (PDF, 159 Kb) | Abstract

*φ*(

*n*) +

*d*(

*n*) =

*n*, and prove related new inequalities.

**Equalities between greatest common divisors involving three coprime pairs**

*Original research paper. Pages 5—7*

Rogelio Tomás García

Full paper (PDF, 140 Kb) | Abstract

*a*and

_{i}*b*positive integers such that gcd(

_{i}*a*) = 1 for

_{i}, b_{i}*i*∈ {1, 2, 3} and

*d*= |

_{ij}*a*|, then gcd(

_{i}b_{j}− a_{j}b_{i}*d*

_{32};

*d*

_{31}) = gcd(

*d*

_{32};

*d*

_{21}) = gcd(

*d*

_{31};

*d*

_{21}): The proof uses properties of Farey sequences.

**Some modular considerations regarding odd perfect numbers – Part II**

*Original research paper. Pages 8—24*

Jose Arnaldo Bebita Dris and Immanuel Tobias San Diego

Full paper (PDF, 181 Kb) | Abstract

*p*and

*k*modulo 16, and show conditions under which the respective congruence classes for

*σ*(

*m*

^{2}) (modulo 8) are attained, if

*p*

^{k}m^{2}is an odd perfect number with special prime

*p*. We prove that

*σ*(*m*^{2}) ≡ 1 (mod 8) holds only if*p*+*k*≡ 2 (mod 16).*σ*(*m*^{2}) ≡ 3 (mod 8) holds only if*p*−*k*≡ 4 (mod 16).*σ*(*m*^{2}) ≡ 5 (mod 8) holds only if*p*+*k*≡ 10 (mod 16).*σ*(*m*^{2}) ≡ 7 (mod 8) holds only if*p*−*k*≡ 4 (mod 16).

We express gcd(*m*^{2}; *σ*(*m*^{2})) as a linear combination of *m*^{2} and *σ*(*m*^{2}). We also consider some applications under the assumption that *σ*(*m*^{2}) / *p ^{k}* is a square. Lastly, we prove a last-minute conjecture under this hypothesis.

**On the quantity I(q^{k}) + I(n^{2}) where q^{k} n^{2} is an odd perfect number**

*Original research paper. Pages 25—32*

Jose Arnaldo Bebita Dris

Full paper (PDF, 194 Kb) | Abstract

*I*(

*q*) +

^{k}*I*(

*n*

^{2}), where

*q*

^{k}n^{2}is an odd perfect number with special prime

*q*and

*I*(

*x*) is the abundancy index of the positive integer

*x*.

**Bi-unitary multiperfect numbers, III**

*Original research paper. Pages 33—67*

Pentti Haukkanen and Varanasi Sitaramaiah

Full paper (PDF, 319 Kb) | Abstract

*d*of a positive integer

*n*is called a unitary divisor if gcd(

*d*,

*n/d*) = 1, and

*d*is called a bi-unitary divisor of

*n*if the greatest common unitary divisor of

*d*and

*n/d*is unity. The concept of a bi-unitary divisor is due to D. Surynarayana (1972). Let

*σ*

^{∗∗}(

*n*) denote the sum of the bi-unitary divisors of

*n*. A positive integer

*n*is called a bi-unitary multiperfect number if

*σ*

^{∗∗}(

*n*) =

*kn*for some

*k*≥ 3. For

*k*= 3 we obtain the bi-unitary triperfect numbers.

Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers. The present paper is part III in a series of papers on even bi-unitary multiperfect numbers. In parts I and II we found all bi-unitary triperfect numbers of the form

*n*= 2

*, where 1 ≤*

^{a}u*a*≤ 5 and

*u*is odd. There exist exactly six such numbers. In this part we examine the case

*a*= 6. We prove that if

*n*= 2

^{6}

*u*is a bi-unitary triperfect number, then

*n*= 22848,

*n*= 342720,

*n*= 51979200 or

*n*= 779688000.

**On quasimultiperfect numbers**

*Original research paper. Pages 68—73*

P. Anantha Reddy, C. Sunitha and V. Siva Rama Prasad

Full paper (PDF, 167 Kb) | Abstract

*n*, let

*σ*(

*n*) and

*ω*(

*n*) respectively denote the sum of the positive divisors of

*n*and the number of distinct prime factors of

*n*. A positive integer

*n*is called a quasimultiperfect (QM) number if

*σ*(

*n*) =

*kn*+ 1 for some integer

*k*≥ 2. In this paper we give some necessary conditions to be satisfied by the prime factors of QM number

*n*with

*ω*(

*n*) = 3 and

*ω*(

*n*) = 4. Also we show that no QM

*n*with

*ω*(

*n*) = 4 can be a fourth power of an integer.

**Notes on the Hermite-based poly-Euler polynomials with a q-parameter**

*Original research paper. Pages 74—82*

Burak Kurt

Full paper (PDF, 206 Kb) | Abstract

*q*-parameter. We give some basic properties and identities for these polynomials. Furthermore, we prove two explicit relations.

**Bivariate Mersenne polynomials and matrices**

*Original research paper. Pages 83—95*

Francisco Regis Vieira Alves

Full paper (PDF, 254 Kb) | Abstract

**Half self-convolution of the k-Fibonacci sequence**

*Original research paper. Pages 96—106*

Sergio Falcon

Full paper (PDF, 202 Kb) | Abstract

*k*-Fibonacci numbers

*F*and

_{k,i}*F*are equidistant if

_{k,j}*j*=

*n*−

*i*and then we study some properties of these pairs of numbers. As a main result, we look for the formula to find the generating function of the product of the equidistant numbers, their sums and their binomial transforms. Next we apply this formula to some simple cases but more common than the general. In particular, we define the half self-convolution of the

*k*-Fibonacci and

*k*-Lucas sequences. Finally, we study the sum of these new sequences, their recurrence relations, and their generating functions.

**A single parameter Hermite–Padé series representation for Apéry’s constant**

*Original research paper. Pages 107—134*

Anier Soria-Lorente and Stefan Berres

Full paper (PDF, 285 Kb) | Abstract

*ζ*(3) which only depends on one single integer parameter. This is accomplished by deducing a Hermite–Padé approximation problem using ideas of Sorokin (1998). As a consequence we get a new recurrence relation for the approximation of

*ζ*(3) as well as a corresponding new continued fraction expansion for

*ζ*(3), which do no reproduce Apéry’s phenomenon, i.e., though the approaches are different, they lead to the same sequence of Diophantine approximations to

*ζ*(3). Finally, the convergence rates of several series representations of

*ζ*(3) are compared.

**A generalization to almost balancing and cobalancing numbers using triangular numbers**

*Original research paper. Pages 135—148*

S. G. Rayaguru and G. K. Panda

Full paper (PDF, 206 Kb) | Abstract

numbers. In case of almost balancing numbers, this difference is kept 1; which is the first triangular number. Some specific representations of these numbers in terms of balancing and balancing related numbers are established and few more results with triangular, square triangular, balancing and balancing related numbers are also studied so as to generalize the identities obtained by A. Tekcan.

**Markov equation with components of some binary recurrent sequences**

*Original research paper. Pages 149—159*

S. G. Rayaguru, M. K. Sahukar and G. K. Panda

Full paper (PDF, 211 Kb) | Abstract

*U*}

_{n}_{n≥0}is defined by

*U*

_{n+1}=

*rU*+

_{n}*sU*

_{n−1};

*n*≥ 0 with

*U*

_{0}= 0;

*U*

_{1}= 1 of which the Fibonacci sequence (

*F*) is the particular case

_{n}*r*=

*s*= 1. In 2018, F. Luca and A. Srinivasan searched for the solutions

*x, y, z*∈

*F*of the Markov equation

_{n}*x*

^{2}+

*y*

^{2}+

*z*

^{2}= 3

*xyz*and proved that (

*F*

_{1};

*F*

_{2n−1},

*F*

_{2n+1});

*n*≥ 1 is the only solution. In this paper, we extend this work from the Fibonacci sequence to any generalized Lucas sequence

*U*for the case

_{n}*s*= ±1.

**An identity involving Bernoulli numbers and the Stirling numbers of the second kind**

*Original research paper. Pages 160—162*

Sumit Kumar Jha

Full paper (PDF, 123 Kb) | Abstract

*B*denote the Bernoulli numbers, and

_{n}*S*(

*n, k*) denote the Stirling numbers of the second kind. We prove the following identity

To the best of our knowledge, the identity is new.

**On bicomplex generalized Tetranacci quaternions**

*Original research paper. Pages 163—175*

Yüksel Soykan and Erkan Taşdemir

Full paper (PDF, 173 Kb) | Abstract

**Hyperbolic k-Fibonacci and k-Lucas octonions**

*Original research paper. Pages 176—188*

A. D. Godase

Full paper (PDF, 182 Kb) | Abstract

*k*-Fibonacci and

*k*-Lucas octonions. We present Binet’s formulas, Catalan’s identity, Cassini’s identity, d’Ocagne’s identity and generating functions for the

*k*-Fibonacci and

*k*-Lucas hyperbolic octonions.

**Identities on generalized Fibonacci and Lucas numbers**

*Original research paper. Pages 189—202*

K. M. Nagaraja and P. Dhanya

Full paper (PDF, 205 Kb) | Abstract

**Fibonacci–Lucas identities and the generalized Trudi formula**

*Original research paper. Pages 203—217*

Taras Goy and Mark Shattuck

Full paper (PDF, 232 Kb) | Abstract

**On intercalated Fibonacci sequences**

*Original research paper. Pages 218—223*

Krassimir T. Atanassov and Anthony G. Shannon

Full paper (PDF, 81 Kb) | Abstract

pulsated sequences described at previous Fibonacci conferences. We relate these sequences to the sequence {

*y*

_{n}}

_{n ≥ 0}= {0, 1, 4, 15, 56,…}.

**On the Padovan p-circulant numbers**

*Original research paper. Pages 224—233*

Güzel İpek, Omür Deveci and Anthony G. Shannon

Full paper (PDF, 159 Kb) | Abstract

*p*-circulant numbers by using circulant matrices which are obtained from the characteristic polynomials of the Padovan p-numbers. Then, we derive the permanental and the determinantal representations of the Padovan

*p*-circulant numbers by using certain matrices which are obtained from the generating matrix of Padovan

*p*-circulant sequence. Also, we obtain the combinatorial representation, the exponential representation and the sums of the Padovan

*p*-circulant numbers by the aid of the generating function and the generating matrix of the Padovan

*p*-circulant sequence.

**A refinement of the 3 x + 1 conjecture**

*Original research paper. Pages 234—244*

Roger Zarnowski

Full paper (PDF, 265 Kb) | Abstract

*x*+ 1 conjecture pertains to iteration of the function

*T*defined by

*T*(

*x*) =

*x*/2 if

*x*is even and

*T*(

*x*) = (3

*x*+ 1)/2 if

*x*is odd. The conjecture asserts that the trajectory of every positive integer eventually reaches the cycle (2, 1). We show that the essential dynamics of

*T*-trajectories can be more clearly understood by restricting attention to numbers congruent to 2 (mod 3). This approach leads to an equivalent conjecture for an underlying function

*T*whose iterates eliminate many extraneous features of T-trajectories. We show that the function

_{R}*T*that governs the refined conjecture has particularly simple mapping properties in terms of partitions of the set of integers, properties that have no parallel in the classical formulation of the conjecture. We then use those properties to obtain a new characterization of

_{R}*T*-trajectories and we show that the dynamics of the 3

*x*+ 1 problem can be reduced to an iteration involving only numberscongruent to 2 or 8 (mod 9).

**On the constant congruence speed of tetration**

*Original research paper. Pages 245—260*

Marco Ripà

Full paper (PDF, 265 Kb) | Abstract

*for*

^{b}a*a*∈ ℕ − {0, 1}, is characterized by fascinating periodicity properties involving its rightmost figures, in any numeral system. Taking into account a radix-10 number system, in the book “La strana coda della serie

*n*^

*n*^ … ^

*n*” (2011), the author analyzed how many new stable digits are generated by every unitary increment of the hyperexponent

*b*, and he indicated this value as

*V*(

*a*) or “congruence speed” of

*a*≢ 0 (mod 10). A few conjectures about

*V*(

*a*) arose. If

*b*is sufficiently large, the congruence speed does not depend on

*b*, taking on a (strictly positive) unique value. We derive the formula that describes

*V*(

*a*) for every

*a*ending in 5. Moreover, we claim that

*V*(

*a*) = 1 for any

*a*(mod 25) ∈ {2, 3, 4, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23} and

*V*(

*a*) ≥ 2 otherwise. Finally, we show the size of the fundamental period

*P*for any of the remaining values of the congruence speed: if

*V*(

*a*) ≥ 2, then

*P*(

*V*(

*a*)) = 10

^{V(a)+1}.

*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-NP1-15/2019.*