MathDB

Let

f : N \to N

such that

f(p) = 1

for all p prime and

f(ab) =bf(a) + af(b)

for all

a, b \in N

. Prove that if

n = p^{a_1}_1 p^{a_1}_2... p^{a_1}_k

is the canonical distribution of

n

and

p_i

does not divide

a_i

(

i = 1, 2, ..., k

) then

\frac{n}{gcd(n,f(n))}

is square free (not divisible by a square greater than

1

Problem Statement