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MULTIPLICATIVE FUNCTIONS k-ADDITIVE ON GENERALISED OCTAGONAL NUMBERS
Published online by Cambridge University Press: 27 August 2024
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.
MSC classification
Information
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
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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