Formula for the sum of distinct prime factors of $n$

Question

This seems like a simple question, but I have been searching the internet forever to find an answer. Is there a formula for the sum of the distinct prime factors of a given positive integer $n$?

Answer 1

There doesn't seem to be a direct formula for the sum of distinct prime factors of n. If n is prime the answer is n itself. If n is composite and large, then first we have to find the prime decomposition of n, which is already difficult enough (for the number of primes less than an integer n see: https://en.wikipedia.org/wiki/Prime-counting_function). If $n=\prod_{i}p_{i}^{a_{i}}$ then for the sum of distinct prime factors of n we can write $f(n)=\sum_{k}p_{{i}}$.

Answer 2

This is closely related to the Euler totient function.