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MULTIPLICATIVE FUNCTIONS k-ADDITIVE ON GENERALISED OCTAGONAL NUMBERS

Part of: Elementary number theory Sequences and sets

Published online by Cambridge University Press:  27 August 2024

ELCHIN HASANALIZADE Open the ORCID record for ELCHIN HASANALIZADE [Opens in a new window]  and
POO-SUNG PARK Open the ORCID record for POO-SUNG PARK [Opens in a new window]
ELCHIN HASANALIZADE*
Affiliation:
School of IT and Engineering, ADA University, Ahmadbey Aghaoghlu str. 61, Baku AZ1008, Azerbaijan
POO-SUNG PARK
Affiliation:
Department of Mathematics Education, Kyungnam University, Changwon 51767, Republic of Korea e-mail: pspark@kyungnam.ac.kr
*
e-mail: ehasanalizade@ada.edu.az
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Abstract

Let $k\geq 4$ be an integer. We prove that the set $\mathcal {O}$ of all nonzero generalised octagonal numbers is a k-additive uniqueness set for the set of multiplicative functions. That is, if a multiplicative function $f_k$ satisfies the condition

$$ \begin{align*} f_k(x_1+x_2+\cdots+x_k)=f_k(x_1)+f_k(x_2)+\cdots+f_k(x_k) \end{align*} $$

for arbitrary $x_1,\ldots ,x_k\in \mathcal {O}$, then $f_k$ is the identity function $f_k(n)=n$ for all $n\in \mathbb {N}$. We also show that $f_2$ and $f_3$ are not determined uniquely.

Keywords

additive uniquenessmultiplicative functionsgeneralised octagonal numbers

MSC classification

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas
Secondary: 11B99: None of the above, but in this section

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 111 , Issue 2 , April 2025 , pp. 212 - 222
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This research was supported by ADA University Faculty Research and Development Funds and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2021R1A2C1092930).

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