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When I graph the sum of each number in the prime factorization of n, I get a strange graph. The individual values seem random, but it definitely has a pattern. Do we know why this is, and if so, why?

To be clear, I'm summing like this:

$f(36) = (2 + 2 + 3 + 3) = 10$

rather than

$f(36) = (2^2 + 3^2) = 13$

Sum of prime factorization

furthermore, not only does it have an overall linear slope, it appears to have (at least) 3 smaller lines, highlighted here:

Sum of prime factorization with more lines

I'm guessing that we don't know exactly why, but even then, might there be some vague hints that we've discovered, at least?

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  • $\begingroup$ is your second example correct? "rather than $f(36)=(2^2+3^3)=15$"? $\endgroup$ – iadvd Sep 14 '15 at 4:32
  • $\begingroup$ @iadvd Whup, typo. Meant 13. $\endgroup$ – AlphaModder Sep 14 '15 at 4:33
  • $\begingroup$ $13$ and $3^2$ not $3^3$ :) $\endgroup$ – iadvd Sep 14 '15 at 4:33
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    $\begingroup$ I would venture to say that the overall linear shape of the data stems from the fact that $f(p)=p$ for prime $p$. That certainly covers the primes. Then the $f(pq)$ is linear in prime $q$ for constant $p$. See where this is going? $\endgroup$ – Terra Hyde Sep 14 '15 at 4:34
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    $\begingroup$ That also accounts for the layering, by the way. $\endgroup$ – Terra Hyde Sep 14 '15 at 4:36
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Notice that for $p\in\mathbb{P}$, $f(p)=p$, so the primes map linearly with slope $1$.

Then let $k\in\mathbb{Z}^+$ and consider: $$f(kp)=f(k)+f(p)=f(k)+p$$ The slope isn't readily obvious for this one, though, so notice that $$k\cdot(p+m)=kp+km$$ and that $$f(k\cdot(p+m))=f(k)+p+m$$ whenever $p+m\in\mathbb{P}$. So the slope will be $\frac{m}{km}=\frac1k$ between $kp$ and $k\cdot(p+m)$. Since the slope is independent of $p$ or $m$, you have the slope for the line related to $k$.

The pattern arises simply because the function sort of "plays nicely" with the primes.

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    $\begingroup$ In other words, the second line (of slope $\frac12$) is from numbers of the form $2p$; the third line (of slope $\frac13$) is from numbers of the form $3p$; and so on. $\endgroup$ – Greg Martin Sep 14 '15 at 5:44
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http://timeblimp.com/?page_id=1194 First Line: Primes

Second Line: 2 times Primes

Third Line : 3 times Primes

Fourth line: 2 times 3 times Primes ect

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