As I said, the key to understanding the Bous. primes is A330545 - they occur whenn A330545 is 0. Walter's A282178 primes occur whenn 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 On Fri, Dec 20, 2019 at 6:22 AM Walter Trump <w@trump.de> wrote:
There exists another "Boustrophedon"-sequence in OEIS.
I like the name "Boustrophedon primes" and appreciate all the work that was done in a short time for A330339. What I do not like is that Boustrophedon primes can only occur on the left ends of the lines and never on the right ends (except of the prime 2). Therefore I thought about a slightly different definition and found another "Boustrophedon"-sequence which already exists in OEIS. In the following graphic all primes are represented by colored dots and all other natural numbers by small black dots. https://www.trump.de/definition-A330339-A282178.gif
Neil already added in the description of A282178 a link to A330339. A big b-file with the first 846 elements already exists. A282178 was created by Samuel B. Reid in 2017. A282178 starts with number 1 and has a higher density than A330339.
Please have a look at 3 pictures demonstrating A282178. !!! Zoom in to read all primes !!! https://www.trump.de/A282178-01.gif https://www.trump.de/A282178-02.gif Especially I like the definition using a perfect zigzag path of the natural numbers with primes in the vertices of the path: https://www.trump.de/A282178-zigzag.gif
Walter
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