The most important arithmetic functions are the additive and the multiplicative ones.
An important operation on arithmetic functions is the Dirichlet convolution.
The articles on additive and multiplicative functions contain several examples of arithmetic functions. Here are some examples that are neither additive nor multiplicative:
Examples
1 = 12+02+02+02 = 02+12+02+02 = 02+02+12+02 = 02+02+02+12,