For any given integer $n$, we prime factorize it as follows
$$n = p_1^{k_1} \cdot p_2^{k_2} \cdots p_r^{k_r}. $$
Let $g = \gcd(k_1, k_2, \ldots, k_r)$ and $m_i = k_i / g$.
The function $F$ is defined as: $$F(n) = p_1^{m_1} · p_2^{m_2} \cdots p_r^{m_r}.$$
I need the value of $F(2) + F(3) + \cdots + F(n)$. How shall I proceed?
Example:
$1936 = 2^4.11^2$
g = hcf(4, 2) = 2
$F(1936) = 2^{4/2}.11^{2/2}$
$F(1936) = 2^{2}.11^{1}$
$F(1936) = 44$
Finally, I converged to the following results. $F(n)$ is the smallest number $k$, that can be raised to some power to $\alpha$.
$$F(n) = k^{\alpha}, $$ where k is as small as possible. Now 1 <= $\alpha$ <= $log_2$(n). So we iterate for different values of $\alpha$, and check whether $n^{1/\alpha}$ is integer or not.
This gives us the value of $F(n)$, but I am not able to find the pattern in $F(2) + F(3) + ... + F(n).$