A note on balanced numbers

József Sándor and Krassimir T. Atanassov
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 4, Pages 8–15
DOI: 10.7546/nntdm.2019.25.4.8-15
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Authors and affiliations

József Sándor
Department of Mathematics, Babes–Bolyai University
Str. Kogalniceanu 1, 400084 Cluj-Napoca, Romania

Krassimir T. Atanassov
Department of Bioinformatics and Mathematical Modelling
IBPhBME – Bulgarian Academy of Sciences,
Acad. G. Bonchev Str. Bl. 105, Sofia-1113, Bulgaria

Abstract

A new proof of solvability of equations

    \[\frac{\sigma(n)}{d(n)} = \frac{n}{2}\]

and

    \[\frac{\sigma_k(n)}{d(n)} = \frac{n^k}{2},\]

for k >1 are given. Connections with related problems and inequalities are pointed out, too.

Keywords

  • Arithmetic function
  • Balanced number
  • Inequality
  • Sum of divisors of a number

2010 Mathematics Subject Classification

  • 11A25
  • 26D15

References

  1. Sandor, J. (1990). An application of the Jensen–Hadamard inequality, Nieuw Arch. Wiskunde, 4 (8), 63–66.
  2. Sandor, J., Mitrinovic, D. S. & Crstici, B. (2006). Handbook of Number Theory, Vol. 1,
    Springer.
  3. Sandor, J. (2008). On equation \sigma_k(n)/d(n)=n^k/2, Octogon Math. Mag., 16 (1), 288–290.
  4. Sandor, J. & Kovacs, L. (2008). A note on the arithmetical functions d(n) and \sigma(n), Octogon Math. Mag., 16 (1), 270–274.
  5. Sandor, J. (2009). A double-inequality for \sigma_k(n), Octogon Math. Mag., 17 (1), 285–287.
  6. Sandor, J. (2009). A better lower bound for \sigma_k(n), Octogon Math. Mag., 17 (2), 767–768.
  7. Sandor, J. & Kovacs, L. (2009). An inequality for the number of divisors of n, Octogon Math. Mag., 17 (2), 746–750.
  8. Sandor, J. (2014). An arithmetic inequality, An. St. Univ. Ovidius Constanta, 22 (1), 257–261.
  9. Subbarao, M. V. (1963). Balanced numbers, Solution of Problem E558, Amer. Math. Monthly, 70, 1009–1010.
  10. Vassilev-Missana, M., & Atanassov, K. (2007). A new point of view on perfect and other similar numbers, Advanced Studies on Contemporary Mathematics, 15 (2), 153–169.

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Cite this paper

APA

Sándor, J. & Atanassov, K. (2019). A note on balanced numbers. Notes on Number Theory and Discrete Mathematics, 25(4), 8-15, doi: 10.7546/nntdm.2019.25.4.8-15.

Chicago

Sándor, József and Krassimir Atanassov. (2019). “A Note on Balanced Numbers.” Notes on Number Theory and Discrete Mathematics. Notes on Number Theory and Discrete Mathematics 25, no. 4 (2019): 8-15, doi: 10.7546/nntdm.2019.25.4.8-15.

MLA

Sándor, József and Krassimir Atanassov. (2019). “A Note on Balanced Numbers.” Notes on Number Theory and Discrete Mathematics 25.4 (2019): 8-15. Print, doi: 10.7546/nntdm.2019.25.4.8-15.

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