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I was idly thinking today about the functions $\displaystyle f(n) = \sum_{p \mid n} p$ and $\displaystyle F(n) = \sum_{p^e \| n} ep$, respectively the "sum of prime divisors" function and the "sum of prime divisors with multiplicity" function. These are both additive functions, and seem like fairly natural functions, fitting into the analogies $\omega:\tau::f:\sigma$ and $\Omega:\tau::F:\sigma$, where $\omega$ is the number of prime divisors, $\Omega$ is the number of prime divisors with multiplicity, $\tau$ is the number of divisors, and $\sigma$ is the sum of divisors. For amusement, I was attempting to compute various statistics of these functions, e.g. their averages and their normal order.

I figure these functions must have showed up before, but searching for "sum of prime divisors function" on this site only turned up questions about the sum of divisors function $\sigma$. So, my questions are:

  1. Have these functions been investigated before? Do they have standard names like $\omega$, $\Omega$, $\tau$, and $\sigma$ do?
  2. Is the normal order of these functions know? That is, is there an analog of the Hardy-Ramanujan theorem for these functions? I attempted to compute normal orders myself by applying the Turán-Kubilius inequality to these functions, but I am not an analytic number theorist practiced in such things, and the computation vexed me.
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  • $\begingroup$ See OEIS Entry A008472 for the sum of the distinct primes dividing $n$. $\endgroup$ Commented Nov 26 at 21:48
  • $\begingroup$ So is $F(4)=4$ (since $2^2$ is a divisor of $4$) or is $F(4)=6$ (since $2^1$ and $2^2$ are both divisors of $4$)? $\endgroup$ Commented Nov 26 at 21:53
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    $\begingroup$ OEIS Entry A001414 which gives the sum of primes dividing n (with repetition) is also possibly related. A008472 and A001414 are sometimes referred to as $sopf(n)$ and $sopfr(n)$ respectively. $\endgroup$ Commented Nov 27 at 0:14
  • $\begingroup$ @StevenClark $F(4) = 2 \cdot 2$ since 2 divides 4 with multiplicity 2. Similarly, $F(9) = 6$. The OEIS page for $sopfr(n)$ references this paper by Alladi and Erdor, so I suppose that answers my question of whether these functions have been studied before. $\endgroup$ Commented Nov 27 at 5:52
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    $\begingroup$ Moreover, the mentioned result is a content of ex.265 of Tenenbaum's "Introduction to analytic and probabilistic number theory" (2015), which is a great source for related topics. For further study of normal orders, consult 2 volumes of Elliott's "Probabilistic number theory" $\endgroup$ Commented Nov 28 at 11:43

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I think the comments have satisfactorily answered my original question, so I'll just write up an answer to take this out of the unanswered queue. It seems that these functions were in fact considered by Alladi and Erdős in this paper and a followup. These functions don't seem to have a special name, but Alladi and Erdős call $A*$ what I called $f$, and call $A$ what I call $F$. They compute the average value to be $\displaystyle \frac{\zeta(2) n}{2\log n}$, but they also show that these functions don't have a normal order. The argument for the latter is reproduced in Tenenbaum Introduction to analytic and probabilistic number theory Chapter III.3 exercise 2. Alladi and Erdős comment explicitly that these functions haven't been studied much, but give a few references to prior work.

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