Prime Factorization and Number Theory

Question

Prime factorization of $n$ is $$n = p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$

Let $f(n) = p_1^{e_1}p_2^{e_2}p_3^{e_3}\cdots p_k^{e_k}$

where $e_k=a_k$ if $p_k|a_k$, else $e_k=a_k-1$

I want to find the value of $$S(N) = \sum_{n=2}^{N}f(n)$$

For example if $N=10$ the answer is: $1+1+4+1+1+1+4+3+1=17$

Can I find the value without factorizing all the numbers? For squarefree numbers $f(n)$ is always $1$.

Answer 1

It seems that the problem of determining if $N$ is squarefree isn't easier than factoring $N$. See Detecting squarefree numbers by Andrew Booker. Quote:

For larger $N$ [$>10^{70}$], finding the complete factorization remains the only viable option.