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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 $n$?

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  • $\begingroup$ If $f(n):=\sum_{p|n}p$ where $p$ is taken amongst the prime numbers, then $e^f$ is multiplicative, it can help. $\endgroup$ – Tuvasbien Apr 19 '20 at 17:29
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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 $f(n)=\sum_{k}p_{{i}}$.

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This is closely related to the Euler totient function.

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