Re: [math-fun] OEIS-inspired research project for an undergrad
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). Â --Steve
Since there are still mysteries here, I'm taking the liberty of adding Steve's two plots to A001414. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com 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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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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participants (3)
-
Allan Wechsler -
Neil Sloane -
Steve Witham