On sums of prime factors

TL;DR

This paper derives the asymptotic behavior of the sum of prime factors (with and without repetition) for integers up to x, confirming a previously stated formula and providing new insights into these arithmetic functions.

Contribution

The paper establishes the asymptotic formula for the sum of prime factors, both with and without repetition, extending prior results and confirming earlier conjectures.

Findings

Asymptotic formula for sum of prime factors with repetition.

Asymptotic formula for sum of distinct prime factors.

Confirmation of previously stated conjecture by Jakimcyuk.

Abstract

We study the arithmetic function sopfr$(n)$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$. In particular we obtain the asymptotic formula $$ \sum_{n \leq x} \rm{sopfr}(n) \sim \frac{\pi^2}{12} \frac{x^2}{\log x},$$ which holds as well for the function sopf$(n)$ (OEIS A008472) that just gives the sum of distinct prime factors of $n$. This asymptotic formula was already stated by R. Jakimcyuk \cite{rj12} which was brought to our attention after the completion of the first version of this manuscript.