On some new arithmetic functions involving prime divisors and perfect powers.

Journal article

Bagdasar, Ovidiu and Tatt, Ralph-Joseph 2018. On some new arithmetic functions involving prime divisors and perfect powers. Electronic Notes in Discrete Mathematics. https://doi.org/10.1016/j.endm.2018.11.002
AuthorsBagdasar, Ovidiu and Tatt, Ralph-Joseph
Abstract

Integer division and perfect powers play a central role in numerous mathematical results, especially in number theory. Classical examples involve perfect squares like in Pythagora’s theorem, or higher perfect powers as the conjectures of Fermat (solved in 1994 by A. Wiles [8]) or Catalan (solved in 2002 by P. Mih˘ailescu [4]). The purpose of this paper is two-fold. First, we present some new integer sequences a(n), counting the positive integers smaller than n, having a maximal prime factor. We introduce an arithmetic function counting the number of perfect powers i j obtained for 1 ≤ i, j ≤ n. Along with some properties of this function, we present the sequence A303748, which was recently added to the Online Encyclopedia of Integer Sequences (OEIS) [5]. Finally, we discuss some other novel integer sequences.

Integer division and perfect powers play a central role in numerous mathematical
results, especially in number theory. Classical examples involve perfect squares
like in Pythagora’s theorem, or higher perfect powers as the conjectures of Fermat
(solved in 1994 by A. Wiles [8]) or Catalan (solved in 2002 by P. Mih˘ailescu [4]). The
purpose of this paper is two-fold. First, we present some new integer sequences a(n),
counting the positive integers smaller than n, having a maximal prime factor. We
introduce an arithmetic function counting the number of perfect powers i
j obtained for 1 ≤ i, j ≤ n. Along with some properties of this function, we present the
sequence A303748, which was recently added to the Online Encyclopedia of Integer
Sequences (OEIS) [5]. Finally, we discuss some other novel integer sequences.

KeywordsArithmetic functions; Perfect powers; Integer sequences
Year2018
JournalElectronic Notes in Discrete Mathematics
PublisherElsevier
ISSN1571-0653
Digital Object Identifier (DOI)https://doi.org/10.1016/j.endm.2018.11.002
Web address (URL)http://hdl.handle.net/10545/623232
http://creativecommons.org/licenses/by/4.0/
hdl:10545/623232
Publication dates06 Dec 2018
Publication process dates
Deposited17 Dec 2018, 14:23
SeriesProceedings of TCDM'18
ContributorsUniversity of Derby
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Bagdasar, O., Birluţiu, A., Chen, M. and Popa, I.-L. 2017. Qualitative case study methodology: Automatic design and correction of ceramic colors. IEEE. https://doi.org/10.1109/icstcc.2017.8107118
New results and conjectures on 2-partitions of multisets
Bagdasar, O. and Andrica, D. 2017. New results and conjectures on 2-partitions of multisets. 7th International Conference on Modeling, Simulation, and Applied Optimization, ICMSAO 2017. IEEE Computer Society. https://doi.org/10.1109/icmsao.2017.7934928
ON THE MASKED PERIODICITY OF HORADAM SEQUENCES: A GENERATOR-BASED APPROACH
Bagdasar, Ovidiu D., Larcombe, Peter J. and Bagdasar, O. 2017. ON THE MASKED PERIODICITY OF HORADAM SEQUENCES: A GENERATOR-BASED APPROACH. Fibonacci Quarterly.
LOCAL MAXIMIZERS OF GENERALIZED CONVEX VECTOR-VALUED FUNCTIONS
Bagdasar, Ovidiu, Popovici, Nicolae and Bagdasar, O. 2017. LOCAL MAXIMIZERS OF GENERALIZED CONVEX VECTOR-VALUED FUNCTIONS. Journal of Nonlinear and Convex Analysis.
On the number of complex horadam sequences with a fixed period
Bagdasar, Ovidiu and Larcombe, Peter J. 2013. On the number of complex horadam sequences with a fixed period. Fibonacci Quarterly.
On the characterization of periodic complex Horadam sequences
Bagdasar, Ovidiu and Larcombe, Peter J. 2013. On the characterization of periodic complex Horadam sequences. Fibonacci Quarterly.
On a result of Bunder involving Horadam sequences: A proof and generalization
Larcombe, Peter J. and Bagdasar, Ovidiu 2013. On a result of Bunder involving Horadam sequences: A proof and generalization. Fibonacci Quarterly.
Influence discovery in semantic networks: An initial approach
Trovati, M. and Bagdasar, O. 2014. Influence discovery in semantic networks: An initial approach. IEEE. https://doi.org/10.1109/uksim.2014.48
Complementary iInequalities involving the Stolarsky Mean
Bagdasar, Ovidiu 2010. Complementary iInequalities involving the Stolarsky Mean. Journal of Inequalities and Applications. https://doi.org/10.1155/2010/492570
Concise computer mathematics: tutorials on theory and problems
Bagdasar, Ovidiu 2013. Concise computer mathematics: tutorials on theory and problems. Springer.
On the characterization of periodic generalized Horadam sequences
Bagdasar, Ovidiu and Larcombe, Peter J. 2014. On the characterization of periodic generalized Horadam sequences. Journal of Difference Equations and Applications. https://doi.org/10.1080/10236198.2014.891022
Local maximum points of explicitly quasiconvex functions
Bagdasar, Ovidiu and Popovici, Nicolae 2014. Local maximum points of explicitly quasiconvex functions. Optimization Letters. https://doi.org/10.1007/s11590-014-0781-3
A horadam-based pseudo-random number generator
Bagdasar, O.D., Chen, M. and Bagdasar, O. 2014. A horadam-based pseudo-random number generator. IEEE. https://doi.org/10.1109/uksim.2014.55
On the structure of periodic complex Horadam orbits
Bagdasar, Ovidiu D., Larcombe, Peter J., Anjum, Ashiq and Bagdasar, O. 2016. On the structure of periodic complex Horadam orbits. Carpathian Journal of Mathematics.
Characterizations of Generalized Exponential Trichotomies for Linear Discrete-time Systems
Popa, I.-L., Ceauşu, T., Bagdasar, O. and Agarwal, R.P. 2016. Characterizations of Generalized Exponential Trichotomies for Linear Discrete-time Systems. Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica. 27 (2), pp. 153-166. https://doi.org/10.2478/auom-2019-0025
On some functions involving the lcm and gcd of integer tuples
Bagdasar, Ovidiu 2014. On some functions involving the lcm and gcd of integer tuples. Scientific Publications of the State University of Novi Pazar Series A: Applied Mathematics, Informatics and mechanics. https://doi.org/10.5937/SPSUNP1402091B
ON A RESULT OF BUNDER INVOLVING HORADAM SEQUENCES: A NEW PROOF
Larcombe, Peter J., Bagdasar, O. and Fennessey, Eric J. 2014. ON A RESULT OF BUNDER INVOLVING HORADAM SEQUENCES: A NEW PROOF. Fibonacci Quarterly. pp. 175-177.