Abstract
AbstractLet
$k\geq 2$
be an integer. We prove that the 2-automatic sequence of odious numbers
$\mathcal {O}$
is a k-additive uniqueness set for multiplicative functions: if a multiplicative function f satisfies a multivariate Cauchy’s functional equation
$f(x_1+x_2+\cdots +x_k)=f(x_1)+f(x_2)+\cdots +f(x_k)$
for arbitrary
$x_1,\ldots ,x_k\in \mathcal {O}$
, then f is the identity function
$f(n)=n$
for all
$n\in \mathbb {N}$
.
Publisher
Cambridge University Press (CUP)
Reference13 articles.
1. Multiplicative functions which are additive on triangular numbers;Park;Bull. Korean Math. Soc.,2021
2. On multiplicative functions which are additive on sums of primes
3. Multiplicative functions satisfying the equation $f({m}^2+{n}^2)=f({m}^2)+f({n}^2)$;Chung;Math. Slovaca,1996
4. Automatic Sequences
5. On k-additive uniqueness of the set of squares for multiplicative functions