Ah, thank you for the additional images. You are right -- there are additional tantalizing hints of structure, other than the "feathers". I have done about an hour of work on this, and I have the flickerings of a theory. If it is easy for you to do, I would like to see a linear/linear version. My theory has to do with A056239; this is very similar in concept to A001414, but instead of summing the primes themselves, you sum the *indices* of the primes, so for 84 = 2 * 2 * 3 * 7, the value is 1 + 1 + 2 + 4 = 8. It's possible that the "feathers" are formed by classes of integers which share the same A056239 value, so you might try doing a plot where points are assigned colors based on A056239. If nobody else tries it, I will eventually try to write some code to do this. On Sat, Jul 4, 2020 at 3:35 PM Steve Witham <sw@tiac.net> wrote:
From: Allan Wechsler <acwacw@gmail.com> Date: 7/3/20, 5:02 PM
Look at the logarithmic scatterplot of A001414.
Explain the feathery diagonal bands at the bottom edge. [& a couple later posts]
I want to start by saying, patterns in semi-random looking computer plots can always be artifacts of the plotting process. Sometimes not artifacts but real (but boring) phenomena aliased against pixels, bin sizes, etc.
But I think the feathers are real.
Hans Havermann had already done:https://oeis.org/A001414/a001414.png which is more spread out.
My version... http://www.mac-guyver.com/switham/2020/07/OEIS_A001414/A001414_lin_log.png ...has white gaps in the upper black lines that I believe are artifacts.
Havermann's doesn't have the same artifacts, but possibly has others.
The clear upper lines are n (the primes), n/2, n/3, n/4... but there is a dark band at sqrt(n).
This is a log-log plot instead of linear-log: http://www.mac-guyver.com/switham/2020/07/OEIS_A001414/A001414_log_log.png Differently interesting at the lower edge. Higher up, you can see sqrt(n), sqrt(n)/2, maybe sqrt(n)/3, but I can't convince myself there's an n^(1/3).
--Steve
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