Definition of $\rho$.

Let's consider a function $\rho$, acting on the prime decomposition of an integer $n$:

$$\begin{matrix} \rho\colon & \mathbb N_{\geqslant 1}&\to &\mathbb N_{\geqslant 1} \\& n=\displaystyle\prod_p p^{\alpha_p}&\mapsto &\displaystyle\sum_p p\cdot\alpha_p .\end{matrix} $$

For example:

$$\rho(12)=\rho(2^2\times 3)=2\times 2+3=7.$$

Context and various information.

We can easily prove two things about $\rho$:

  • $\forall m,n\in \mathbb N_{\geqslant 1},\quad \rho(mn)=\rho(m)+\rho(n)$;

  • $\rho(n)=n\iff \text{ $n$ is a prime number}.$

We can also prove that

$$\rho(n)\geqslant \frac {2\log n}{\log 2}.$$

Here is an illustration of the sequence $\rho(n)$ for $n\in \{1,\ldots,100\}$. We can see that the born $\frac {2\log 2}{\log n}$ fits well. We can also notice that $\rho(n)=n$ if, and only if, $n$ is a prime number.

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Here is the sequence for $n$ up to $10^3$. We can notice that things are ordered at the top and chaotic on the bottom.

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If we plot the sequence for $n$ up to $10^6$, we will notice that this pattern seems to continue. Which raise my question:

Question. What causes those lines to occur? What are the equations of those lines?

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  • 2
    $\begingroup$ So-called integer logarithm or sopfr($n$) - see OEIS A001414 - the top line has slope $1$ and is made up if primes and $4$ $\endgroup$ – Henry Jun 10 '17 at 16:31
  • 1
    $\begingroup$ The next lines below the top lines will be the values for $2\cdot p$ ($\rho(n) \approx n/2$), $3\cdot p$ ($\rho(n) \approx n/3$) etc. $\endgroup$ – Daniel Fischer Jun 10 '17 at 17:30

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