Boustrophedon primes

Any others? (My school notebook is too small :-)

The "boustrophedon primes" can be found on the "zero critical line". The sequence would start with 37, 53, 89, 113, ... [which is not https://oeis.org/A217561].
Best,
É.
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Update, 22:22, Brussels time, Belgium

This popped up on Math-Fun 2 hours ago, with a wonderful picture computed by Walter T.:

> The next Boustrophedon primes are  3821  and  3989. Here is a continuation of Eric's arrangement of numbers (you cannot read thedigits, they are too small):
(click on the track for a larger image)
(This is now in the OEIS – many thanks Neil!-)

A nice idea from Allan W. (always on Math-Fun)

There is an underlying sequence, which tells what columns successive prime numbers land in. This is the alternating sum of the decremented first differences of the primes, with the sequence starting at 2. Just reading off Eric's picture, I get:

2,2,3,2,5,4,7,6,9,4,5,0,3,2,5,0,....

A330339 is then the primes indexed by occurrences of zero in this alternating sum.


Bravo and thank you Walter, Allan, Neil, Tom, Hans, Alexandre...! (and Lindsey  of course – we miss you !-)
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Update December 19th around noon (Brussels time)

Neil S. on Math-Fun:
> The OEIS entry for Éric Angelini's Boustrophedon primes, A330339, has been greatly expanded, thanks to extensive computations by Hans Havermann (TomDuff's calculation agrees with Hans's, although Hans went further). I added two further sequences, A330545 and especially A330547, which show the connection with the ordinary primes and their alternating sums. Hans also produced a graph of 4*10^8 terms of A330545, which must be the longest ski run in the world. Walter Trump's ski run of length 3989 and  550 turns was drawn in the correct downhill direction, but Hans's have been turned sideways.
The ski run in his biggest picture has 4*10^8 terns and length = prime(4*10^8) which is about 8*10^10.


Hans Havermann (hereunder):
Plot of 4*10^8 terms of A330545, sampled every 1000 terms, points joined.

Hans Havermann (hereunder)
More detailed view of terms of A330545 from 290 million to 310 million, sampled every 10 terms, points joined.



Update, December 22nd, 2019

A private mail from Neil to the bunch:

> Since Eric started this balling rolling on the Math Fun list, not the Seq Fans list, and the two lists don't overlap much, I am just sending this to the people who has joined the discussions.

The two main seqs of primes are A282178 and A330339. I just created A330554 for their union (they are closely related). The new one could use a bigger b-file - Giovanni?

I have also updated both of those entries, explaining the connections;and the underlying sequences A330545 and A330547 (they are essentially identical).

Hans Havermann's amazing graph of 4*10^8 terms for A330545 is at the heart of the matter. It illustrates all of these sequences.  A282178 gives the primes on the y=2 slice through the graph, and A330339 on the y=0 slice.So obviously the two slices are closely related.

The big mystery is, what happens to the graph as n increases? I have a lot of guesses (think coin-tossing; arcsine law; Feller Chapter 3) but no results at all.And in particular, are the two sequences of primes infinite? The graphs make it pretty clear the answer is Yes.

Oh, and A008347 is lurking in the background.  If we understood that then things would be easier.  The trouble is, the estimates for it - even assuming the Riemann Hypothesis - aren't very precise.

We are in deep water here, so any help will be welcomed!

Neil
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Xmas day update #1:

Neil (Dec 20, 2019):
> As I said, the key to understanding the Bous. primes is A330545 - theyoccur when n A330545 is 0.Walter's A282178 primes occur when n A330545 is 2.
It is bit nicer in terms of A330547: if that is -1 we get a Bous. prime (A330399), and if it is +1 we get an A282178 prime.The graphs of A330545 and A330547 (they are essentially the same) are quite wild (take a look at Hans Havermann's plot of 4*10^8 terms of A330545).Here is what seems to be going on:The asymptotic formula for the n-th prime  starts off with   p_n ~ n(log n + log log n -1)How good is this?  Well. if you look at the difference, p_n - n(log n + log log n -1), that is a bit like A330545.  What I mean is, if you try to get an estimate for A330545(n), you end up looking at the difference between two quantities, both of which start out n(log n + log log n -1).
Best regards
Neil
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Xmas day update #2:

Frank S.

I let my laptop spin up for a couple days, and calculated the first 163010
numbers in A282178 : http://traxme.net/Boustrophedon.txt
If you plot the occurrences in log-log histogram like this:


http://traxme.net/b_primes.png
It seems that they occur in exponentially larger clusters, at exponentially
greater height. My guess is that the meandering effects slows down as
primes grow larger.
/f
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End note in French
Je veux bien, comme expliqué ici, que « l'amoncellement des instruments de musique, dont personne ne joue, posés par terre, muets, rejoints bientôt par les tuyaux de l'orgue de Cécile, symbolise la musica instrumentalis, inférieure à la musica humana des personnages terrestres, couronnée par la sublime harmonie des sphères célestes, la musica mundana, absolu de la musique »... mais celle des maths peut aussi provoquer des extases, non ?






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