Arithmetical functions associated with divisibility sequences

Anthony G. Shannon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 2, Pages 328–334
DOI: 10.7546/nntdm.2026.32.2.328-334
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Authors and affiliations

Anthony G. Shannon

Warrane College, University of New South Wales
Kensington, NSW 2033, Australia

Abstract

This note looks at some aspects of divisibility sequences and generalized integers, including so-called Fermatian numbers, and extensions of ideas of Mollie Horadam and Morgan Ward.

Keywords

  • Cauchy product
  • Divisibility sequences
  • Fermatian numbers
  • Generalized integers
  • Hurwitz series
  • Rank of apparition
  • Regular

2020 Mathematics Subject Classification

  • 11B75
  • 11Z05
  • 11B65

References

  1. Carlitz, L. (1954). Congruence properties of the polynomials of Hermite, Laguerre and Legendre. Mathematische Zeitschrift, 59, 474–483.
  2. Carlitz, L., & Moser, L. (1966). On some special factorizations of (1 - x^n)/(1 - x). Canadian Mathematical Bulletin, 9, 421–426.
  3. Hardy, G. H., & Wright, E. M. (1965). An Introduction to the Theory of Numbers (4th edition). Oxford: Clarendon Press.
  4. Hoggatt, Jr., V. E., & Long, C. T. (1973). Divisibility properties of generalized Fibonacci polynomials. The Fibonacci Quarterly, 12(2), 113–120.
  5. Horadam, A. F., Loh, R. P., & Shannon, A. G. (1979). Divisibility properties of some Fibonacci-type sequences. In: Horadam, A. F., & Wallis W. D. (eds.). Combinatorial Mathematics VI. Heidelberg, Springer, pp. 55–64.
  6. Horadam, E. M. (1962). Arithmetical functions associated with the unitary divisors of a generalized integer. The American Mathematical Monthly, 69(3), 196–199.
  7. Horadam, E. M. (1964). Ramanujan’s sum for generalized integers. Duke Mathematical Journal, 31(4), 697–702.
  8. Hurwitz, A. (1923). Über die Komposition der quadratischen Formen. Mathematische Annalen, 88 (1/2), 1–25.
  9. Lehmer, D. H. (1930). An extended theory of Lucas’ functions. Annals of Mathematics, 31(3), 419–448.
  10. Long, C., Cohen, G. L., Langtry, T., & Shannon, A. G. (1993). Arithmetic sequences and second order recurrences. In: Bergum. G. E., Philippou, A. N., & Horadam, A. F. (eds.), Applications of Fibonacci Numbers, Volume 5. Dordrecht: Kluwer, pp. 449–457.
  11. Pierce, T. A. (1916). The numerical factors of the arithmetic form. Annals of Mathematics, 18(2), 53–64.
  12. Shannon, A. G. (2004). Some properties of Fermatian numbers. Notes on Number Theory and Discrete Mathematics, 10(2), 25–33.
  13. Shannon, A. G., Uysal, M., & Özkan, E. (2025). Fermatian row and column sums as a family of generalized integers. Notes on Number Theory and Discrete Mathematics, 31(3), 433–442.
  14. Sloane, N. J. A., & Plouffe, S. (1995). The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press. Available online at: https://oeis.org.
  15. Sullivan, R. P. (1987). Semigroups generated by nilpotent transformations. Journal of Algebra, 110(2), 324–345.
  16. Vorob’ev, N. N. (1961). Fibonacci Numbers. Oxford: Pergamon.
  17. Ward, M. (1936). Divisibility sequences. Bulletin of the American Mathematical Society, 42(12), 843–845.
  18. Williams, H. C. (1972). On a generalization of the Lucas function. Acta Arithmetica, 20(1), 33–51.

Manuscript history

  • Received: 15 April 2026
  • Revised: 15 May 2026
  • Accepted: 2 June 2026
  • Online First: 3 June 2026

Copyright information

Ⓒ 2026 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Related papers

  1. Shannon, A. G., Uysal, M., & Özkan, E. (2025). Fermatian row and column sums as a family of generalized integers. Notes on Number Theory and Discrete Mathematics, 31(3), 433–442.
  2. Shannon, A. G. (2019). Applications of Mollie Horadam’s generalized integers to Fermatian and Fibonacci numbers. Notes on Number Theory and Discrete Mathematics, 25(2), 113–126.
  3. Shannon, A. G. (2004). Some properties of Fermatian numbers. Notes on Number Theory and Discrete Mathematics, 10(2), 25–33.

Cite this paper

Shannon, A. G. (2026). Arithmetical functions associated with divisibility sequences. Notes on Number Theory and Discrete Mathematics, 32(2), 328-334, DOI: 10.7546/nntdm.2026.32.2.328-334.

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