MathDB

Determine all functions

f

defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[*]

(i)

f(n) \neq 0

for at least one

n

; [*]

(ii)

f(x y)=f(x)+f(y)

for every positive integers

x

and

y

; [*]

(iii)

there are infinitely many positive integers

n

such that

f(k)=f(n-k)

for all

k<n

N5