Let rad(n) = $\Pi_{primes, p|n}$ p.

Question

Let $\operatorname{rad}(n) = \displaystyle\prod_{\stackrel{p|n}{p \text{ prime }}}p$ .

I have proven that $\operatorname{rad}(n)$ is a multiplicative arithmetic function.

I have also proven that $F(n) = \sum_{d|n} \operatorname{rad} (d)$ is also multiplicative.

Determine a formula for $F(n)$ in terms of the prime factorization of $n$. What is the value of $F(120)$?

I managed to calculate $F(120)$ naively.

$F(120) = 167$

Answer 1

Hint Since $F$ is multiplicative

$$F(p_1^{\alpha_1}p_2^{\alpha_2}....p_k^{\alpha_k})=F(p_1^{\alpha_1})F(p_2^{\alpha_2})....F(p_k^{\alpha_k})$$

So it suffices to find $F(p^\alpha)$. But this is easy:

$$F(p^\alpha)=f(1)+f(p)+f(p^2)+...+f(p^\alpha) \,.$$