completely multiplicative function
A function f(n) defined on the positive integers is completely multiplicative if f(nm)=f(n)f(m) for all pairs n and m (compare this with multiplicative functions). Three simple examples are f(n)=0, f(n)=1, and f(n)=nc (for a fixed positive value c).
f(n) = f(p1)a1. f(p2)a2. ....f(pk)ak.