Let $n$ is a positive integer.
$n = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the complete prime factorization of $n$.
Let me define a function $f(n)$
$f(n) = p_1^{c_1}p_2^{c_2}\cdots p_k^{c_k}$ where $c_k = e_k - 1$
Example:
$72 = 2^33^2$, so $f(72) = 2^{3-1}3^{2-1} = 2^{2}3^{1}=12$
$144 = 2^43^2$, so $f(144) = 2^{4-1}3^{2-1} = 2^{3}3^{1}=24$
Now let $$F(N) = \sum_{n=2}^N f(n)$$
Example: $F(10) = 1 + 1 + 2 + 1 + 1 + 1 + 4 + 3 + 1 = 15$
Now I want to evaluate $F(N)$ for a fairly large value of $N$, say $10^{12}$. Can I do it without factorizing each number?