Abstract
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.
Publisher
Cambridge University Press (CUP)
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