Re: [math-fun] Boustrophedon primes
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
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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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 On Fri, Dec 20, 2019 at 6:26 PM Neil Sloane <njasloane@gmail.com> wrote:
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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Hello Math-Fun, Numbers that, when divided by one of their digits, have another of their digits as remainder: S = 10,13,19,20,21,23,26,29,30,31,32,39,... Check: 10 = 10*1+0 13 = 4*3+1 19 = 2*9+1 20 = 10*2+0 21 = 10*2+1 23 = 7*3+2 26 = 4*6+2 29 = 3*9+2 30 = 10*3+0 31 = 10*3+1 32 = 10*3+2 39 = 4*9+3 ... It would be nice to see the numbers where this is true, whatever the dividing digit is chosen in them (except 0). We could call them NYE-numbers as 20191231 is such an integer. Check: 20191231 = 10095615*2+1 20191231 = 20191231*1+0 20191231 = 2243470*9+1 20191231 = 6730410*3+1 P.-S. There are also the integers where this routine is true for exactly two distinct digits, like 127 (not 133), as 127 = 63*2+1 and 127 = 18*7+1. Or exactly three distinct digits, four, etc. Best, É. (and forgive me, as usual, if this is old hat or if some typos are still present).
I created A330562: Positive numbers k with property that if d is any nonzero digit of k then k mod d is also a digit of k. 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 101, 102, 103, 104, 105, 109, 110, ... and commented that k must always have a zero digit. Maybe someone could check? I did it in a hurry. On Sat, Dec 28, 2019 at 12:08 PM Éric Angelini <bk263401@skynet.be> wrote:
Hello Math-Fun, Numbers that, when divided by one of their digits, have another of their digits as remainder: S = 10,13,19,20,21,23,26,29,30,31,32,39,... Check: 10 = 10*1+0 13 = 4*3+1 19 = 2*9+1 20 = 10*2+0 21 = 10*2+1 23 = 7*3+2 26 = 4*6+2 29 = 3*9+2 30 = 10*3+0 31 = 10*3+1 32 = 10*3+2 39 = 4*9+3 ... It would be nice to see the numbers where this is true, whatever the dividing digit is chosen in them (except 0). We could call them NYE-numbers as 20191231 is such an integer. Check: 20191231 = 10095615*2+1 20191231 = 20191231*1+0 20191231 = 2243470*9+1 20191231 = 6730410*3+1
P.-S. There are also the integers where this routine is true for exactly two distinct digits, like 127 (not 133), as 127 = 63*2+1 and 127 = 18*7+1. Or exactly three distinct digits, four, etc. Best, É. (and forgive me, as usual, if this is old hat or if some typos are still present).
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Just for grins, primes in A330562 start: 101, 103, 109, 401, 601, 701, 1009, 1021, 1033, 1051, 1063, 1201, 1303, 1801, 2011, 2017, 2081, 2801, 3001, 3061, 3301, 4001, 4021, 4201, 4801, 5011, 5021, 5051, 5081, 5101, 5501, 5801, 6301, ... Done in a hurry, so maybe should be checked. On Tue, Dec 31, 2019 at 9:28 AM Neil Sloane <njasloane@gmail.com> wrote:
I created A330562: Positive numbers k with property that if d is any nonzero digit of k then k mod d is also a digit of k. 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 101, 102, 103, 104, 105, 109, 110, ... and commented that k must always have a zero digit. Maybe someone could check? I did it in a hurry.
On Sat, Dec 28, 2019 at 12:08 PM Éric Angelini <bk263401@skynet.be> wrote:
Hello Math-Fun, Numbers that, when divided by one of their digits, have another of their digits as remainder: S = 10,13,19,20,21,23,26,29,30,31,32,39,... Check: 10 = 10*1+0 13 = 4*3+1 19 = 2*9+1 20 = 10*2+0 21 = 10*2+1 23 = 7*3+2 26 = 4*6+2 29 = 3*9+2 30 = 10*3+0 31 = 10*3+1 32 = 10*3+2 39 = 4*9+3 ... It would be nice to see the numbers where this is true, whatever the dividing digit is chosen in them (except 0). We could call them NYE-numbers as 20191231 is such an integer. Check: 20191231 = 10095615*2+1 20191231 = 20191231*1+0 20191231 = 2243470*9+1 20191231 = 6730410*3+1
P.-S. There are also the integers where this routine is true for exactly two distinct digits, like 127 (not 133), as 127 = 63*2+1 and 127 = 18*7+1. Or exactly three distinct digits, four, etc. Best, É. (and forgive me, as usual, if this is old hat or if some typos are still present).
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I've added Frank Stevenson's remarkable table and graph of 163010 terms to A282178. The red line on the graph shows that the terms to this point roughly satisfy log a(n) <= 3*log n. On Tue, Dec 24, 2019 at 2:59 AM Frank Stevenson < frankstevensonmobile@gmail.com> wrote:
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
On Fri, Dec 20, 2019 at 6:26 PM Neil Sloane <njasloane@gmail.com> wrote:
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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Correction. On Dec 29 I said
I've added Frank Stevenson's remarkable table and graph of 163010 terms to A282178. The red line on the graph shows that the terms to this point roughly satisfy log a(n) <= 3*log n.
I should have said that it seems that a(n) is roughly growing like n^c where c is about 2.7. On Sun, Dec 29, 2019 at 11:11 AM Neil Sloane <njasloane@gmail.com> wrote:
I've added Frank Stevenson's remarkable table and graph of 163010 terms to A282178. The red line on the graph shows that the terms to this point roughly satisfy log a(n) <= 3*log n.
On Tue, Dec 24, 2019 at 2:59 AM Frank Stevenson < frankstevensonmobile@gmail.com> wrote:
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
On Fri, Dec 20, 2019 at 6:26 PM Neil Sloane <njasloane@gmail.com> wrote:
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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participants (5)
-
Frank Stevenson -
James Buddenhagen -
Neil Sloane -
Walter Trump -
Éric Angelini