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# A Note on r-Full Numbers of the Form l m + k

Periodica Mathematica Hungarica
Author: Florian Luca
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# On the equation 1k+2k+···+nk=x 2

Periodica Mathematica Hungarica
Author: Florian Luca
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# On the sum of two divisors of (n 2 + 1)/2

Periodica Mathematica Hungarica
Authors: Andrej Dujella and Florian Luca

## Abstract

In this paper, given a pair of odd coprime integers δ and ɛ, we study the positive n such that (n 2 + 1)/2 has two divisors d 1 and d 2 summing up to δn + ɛ.

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# Common factors of shifted Fibonacci numbers

Periodica Mathematica Hungarica
Authors: Santos Hernández and Florian Luca

## Abstract

For any positive integer n let Fn be the n-th Fibonacci number. Given positive integers a and b, we study the size of the greatest common divisor of Fn + a and Fm + b for varying positive integers m and n.

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# Arithmetic functions monotonic at consecutive arguments

Studia Scientiarum Mathematicarum Hungarica
Authors: Jean-Marie Koninck and Florian Luca

For a large class of arithmetic functions f, it is possible to show that, given an arbitrary integer κ ≤ 2, the string of inequalities f(n + 1) < f(n + 2) < … < f(n + κ) holds for in-finitely many positive integers n. For other arithmetic functions f, such a property fails to hold even for κ = 3. We examine arithmetic functions from both classes. In particular, we show that there are only finitely many values of n satisfying σ2(n − 1) < σ2 < σ2(n + 1), where σ2(n) = ∑d|n d 2. On the other hand, we prove that for the function f(n) := ∑p|n p 2, we do have f(n − 1) < f(n) < f(n + 1) in finitely often.

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# Perfect Pell and Pell–Lucas numbers

Studia Scientiarum Mathematicarum Hungarica
Authors: Jhon J. Bravo and Florian Luca

## Abstract

The Pell sequence $(Pn)n=0∞$ is given by the recurrence Pn = 2Pn −1 + Pn −2 with initial condition P 0 = 0, P 1 = 1 and its associated Pell-Lucas sequence $(Qn)n=0∞$ is given by the same recurrence relation but with initial condition Q 0 = 2, Q 1 = 2. Here we show that 6 is the only perfect number appearing in these sequences. This paper continues a previous work that searched for perfect numbers in the Fibonacci and Lucas sequences.

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# Short interval asymptotics for a class of arithmetic functions

Acta Mathematica Hungarica
Authors: Mübariz Z. Garaev, Florian Luca, and Werner Georg Nowak

## Summary

We provide a general asymptotic formula which permits applications to sums like \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sum_{x< n\le x+y} \big(d(n)\big)^2, \quad \sum_{x< n\le x+y} d(n^3),\quad \sum_{x< n\le x+y}\big(r(n)\big)^2, \quad \sum_{x< n\le x+y}r(n^3),$ \end{document}$where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$d(n)$\end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$r(n)$\end{document} are the usual arithmetic functions (number of divisors, sums of two squares), and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$y$\end{document} is small compared to~\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$x\$ \end{document}.

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# Linear independence results for the reciprocal sums of Fibonacci numbers associated with Dirichlet characters

Studia Scientiarum Mathematicarum Hungarica
Authors: Hiromi Ei, Florian Luca, and Yohei Tachiya

Let {F n}n≥0 be the sequence of Fibonacci numbers. The aim of this paper is to give linear independence results over $ℚ(5)$ for the infinite series $∑n=1∞χj(n)/Fn$ with certain nonprincipal real Dirichlet characters χ j. We also deduce the irrationality results for the special principal Dirichlet characters and for other multiplicative functions.

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