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).

If f(n) is multiplicative and we factor n into distinct primes as n=p1a1. p2a2. ....pkak, then

f(n) = f(p1)a1. f(p2)a2. ....f(pk)ak.

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