MULTIPLICATIVE FUNCTIONS k-ADDITIVE ON GENERALISED OCTAGONAL NUMBERS

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.