# 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*)=*n*^{c}
(for a fixed positive value c).

If f(*n*) is multiplicative and we factor
*n* into distinct primes as
*n*=*p*_{1}^{a1}^{.}
*p*_{2}^{a2}^{.}
...^{.}*p*_{k}^{ak},
then

f(n) = f(p_{1})^{a1}^{.}f(p_{2})^{a2}^{.}...^{.}f(p_{k})^{ak}.

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