[math-fun] Boustrophedon primes
The next Boustrophedon primes are 3821 and 3989. Here is a continuation of Eric's arrangement of numbers (you cannot read the digits, they are too small): https//www.trump.de/Boustrophedon-Primes.png Walter
Thank you Walter! This is an incentive for ski slopes! Best, É.
Le 17 déc. 2019 à 20:05, Walter Trump <w@trump.de> a écrit :
The next Boustrophedon primes are 3821 and 3989. Here is a continuation of Eric's arrangement of numbers (you cannot read the digits, they are too small): https//www.trump.de/Boustrophedon-Primes.png
Walter
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See https://oeis.org/A330339 On Tue, Dec 17, 2019 at 4:24 PM Éric Angelini <eric.angelini@skynet.be> wrote:
Thank you Walter! This is an incentive for ski slopes! Best, É.
Le 17 déc. 2019 à 20:05, Walter Trump <w@trump.de> a écrit :
The next Boustrophedon primes are 3821 and 3989. Here is a continuation of Eric's arrangement of numbers (you cannot read the digits, they are too small): https//www.trump.de/Boustrophedon-Primes.png
Walter
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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. On Tue, Dec 17, 2019 at 5:14 PM Neil Sloane <njasloane@gmail.com> wrote:
On Tue, Dec 17, 2019 at 4:24 PM Éric Angelini <eric.angelini@skynet.be> wrote:
Thank you Walter! This is an incentive for ski slopes! Best, É.
Le 17 déc. 2019 à 20:05, Walter Trump <w@trump.de> a écrit :
The next Boustrophedon primes are 3821 and 3989. Here is a continuation of Eric's arrangement of numbers (you cannot read the digits, they are too small): https//www.trump.de/Boustrophedon-Primes.png
Walter
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Allan, I had exactly the same idea, and this afternoon I created A330545! It is indeed the key to the whole problem - and of course the recurrence makes it a lot easier to compute. On Tue, Dec 17, 2019 at 5:27 PM Allan Wechsler <acwacw@gmail.com> wrote:
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.
On Tue, Dec 17, 2019 at 5:14 PM Neil Sloane <njasloane@gmail.com> wrote:
On Tue, Dec 17, 2019 at 4:24 PM Éric Angelini <eric.angelini@skynet.be> wrote:
Thank you Walter! This is an incentive for ski slopes! Best, É.
Le 17 déc. 2019 à 20:05, Walter Trump <w@trump.de> a écrit :
The next Boustrophedon primes are 3821 and 3989. Here is a continuation of Eric's arrangement of numbers (you cannot read the digits, they are too small): https//www.trump.de/Boustrophedon-Primes.png
Walter
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I wrote a little program and computed the first 2^32 (4294967296) steps of the walk. Unless my program is busted, there are 422 boustrophedon primes up to 4294967296: 37 53 89 113 3821 3989 4657 28661 29021 41641 41669 44249 50909 56053 57041 57301 133981 16501361 46178761 47633441 47633477 47722049 47736121 47774621 47803477 47810209 47835013 47835341 47854969 47862413 47865017 49448573 49448617 49449133 49464241 50143469 50143601 53757637 53757797 53757881 56497153 56527117 56707681 56708429 56713697 56740009 56741309 56808877 56932357 56932801 56976697 57364081 57369617 57488653 57489233 57490997 57493861 57494573 57566141 57567161 57570521 57571009 57577061 57788597 57808649 57811097 57811609 68807029 68812013 68835721 68836129 68836333 68839289 68847853 69985609 69985633 69987377 69996869 69999961 70001873 70012781 70015217 70095209 70095661 70113833 70114181 70144541 70420577 70432081 70448669 70475813 70482637 70484461 70485469 70490569 70503593 70574173 70585693 70645409 70663657 70685837 70709909 71259701 71259893 71261317 71604517 71604529 71914933 72011449 72011809 72048113 72062257 72062489 72067181 72203473 72203513 72270113 72271217 72313321 72326113 72332233 72332257 72333889 72353213 72421493 87851969 87853069 89935973 89937677 89938561 89940953 89941589 89964913 90007633 90011021 90011773 90016897 90019733 90020257 90051109 90051173 90136957 202673453 202866649 202870021 202881389 202897241 202903321 203301173 203301457 203301797 203302381 206443357 206446441 206447893 206449157 206484793 206514577 206531777 206532901 206539037 206539121 206539153 206539373 206539601 206564537 206564681 206602681 206602897 206613877 206614081 206636933 206637133 206642801 206642881 206653009 206665937 207824657 207933617 207964193 207964333 207964513 209274629 209292329 209292437 209293061 209297017 209303509 209304457 209304497 209380229 209390729 209479993 209480189 209499677 209502677 211887857 211912229 211915213 211919237 211940189 212118769 212119549 212124733 212149849 212168933 212169613 212176841 212862173 212997553 213161369 213227897 213239153 390767609 390783053 390802837 390895961 402489449 402492089 402614441 402625793 402635689 402635741 402635777 402642677 402670817 402788173 402795361 402831197 402834869 403173161 403174249 403397653 403401937 403402729 415301401 415301629 415302637 415304941 426013633 426014293 426155537 430961129 430961417 430974949 430979557 430979761 435031369 435032077 435052957 435053261 435090749 435098393 435098501 435098753 435099593 435100933 435132277 436555673 436560977 436626709 436689257 437392957 437522401 437605409 437699849 437703457 437703797 437704781 437764441 437768033 437768257 437808881 437845297 437862037 437877929 451445021 451757717 451757821 451764361 453602309 453612541 453613357 453613649 453667813 453672221 509656897 510610193 510617209 510647861 510647953 510652481 511498313 511500877 511509841 511570597 511570733 512279249 512524093 512524889 512525653 512526017 512590217 512590361 512590373 512592989 512607157 513134021 513146189 513243281 513243641 513248629 513252449 513396893 513559577 513643381 513643573 513649393 515232449 515232833 515232877 515243717 515423497 515423521 515423977 515432017 515432069 515447341 515447473 515453881 515517053 515518777 515574593 515602589 515602973 515612221 515626481 515628901 515642837 515721593 515928137 515930137 515930521 515931061 515939297 663825733 663868973 663870569 663870917 664028941 664029001 664031897 664032001 664040549 664423313 669056321 669080453 669086557 670484557 670502341 670535329 670536137 670542797 670574101 670574813 670585313 670674481 670675157 670681097 671018009 671019593 671019949 671051321 671051357 671084669 671096501 671099909 671104673 671140397 671140469 671141941 671164717 671178817 671179477 671184809 671224033 671225249 671235209 671484277 671501141 671554193 671559269 671564809 671593337 671650961 671650997 671651993 671659229 671667709 672861617 672861949 675572477 675579353 675582877 675674213 682402621 682409177 682669621 682730669 683037629 683049893 683117629 683118673 683152193 683152409 683170613 683176253 683190397 683193529 683193949 683219221 684073009 The gaps between these numbers vary a lot. The record gaps are: n a(n) a(n)-a(n-1) fraction --- -------------- -------------- --------- 1 37 37 1.000000 5 3,821 3,708 0.980110 8 28,661 24,004 0.968180 17 133,981 76,680 0.779431 18 16,501,361 16,367,380 0.998209 19 46,178,761 29,677,400 0.999360 143 202,673,453 112,536,496 0.782962 214 390,767,609 177,528,456 0.860394 423 >4,295,135,754 >3,611,062,745 >0.919011 Here the first column is n, the second is a(n), the third column is the size of the gap between a(n-1) and a(n) and the last column is the fraction of a(n) accounted for by the sum of the preceding record gaps. The last line is a lower bound on the next gap, based on the state of affairs when my program stopped. So the path looks mostly like a very few very long meanders, separated by lots more (relatively) tiny excursions. (But note the 3 record gaps in a row ending at a(17), a(18) and a(19).) I wonder if the path ever returns after this point. I suspect that's a very deep question -- certainly beyond my ken. On Tue, Dec 17, 2019 at 2:14 PM Neil Sloane <njasloane@gmail.com> wrote:
On Tue, Dec 17, 2019 at 4:24 PM Éric Angelini <eric.angelini@skynet.be> wrote:
Thank you Walter! This is an incentive for ski slopes! Best, É.
Le 17 déc. 2019 à 20:05, Walter Trump <w@trump.de> a écrit :
The next Boustrophedon primes are 3821 and 3989. Here is a continuation of Eric's arrangement of numbers (you cannot read the digits, they are too small): https//www.trump.de/Boustrophedon-Primes.png
Walter
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The OEIS entry for Éric Angelini's Boustrophedon primes, A330339, has been greatly expanded, thanks to extensive computations by Hans Havermann (Tom Duff'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. On Wed, Dec 18, 2019 at 7:55 PM Tom Duff <td@pixar.com> wrote:
I wrote a little program and computed the first 2^32 (4294967296) steps of the walk. Unless my program is busted, there are 422 boustrophedon primes up to 4294967296: 37 53 89 113 3821 3989 4657 28661 29021 41641 41669 44249 50909 56053 57041 57301 133981 16501361 46178761 47633441 47633477 47722049 47736121 47774621 47803477 47810209 47835013 47835341 47854969 47862413 47865017 49448573 49448617 49449133 49464241 50143469 50143601 53757637 53757797 53757881 56497153 56527117 56707681 56708429 56713697 56740009 56741309 56808877 56932357 56932801 56976697 57364081 57369617 57488653 57489233 57490997 57493861 57494573 57566141 57567161 57570521 57571009 57577061 57788597 57808649 57811097 57811609 68807029 68812013 68835721 68836129 68836333 68839289 68847853 69985609 69985633 69987377 69996869 69999961 70001873 70012781 70015217 70095209 70095661 70113833 70114181 70144541 70420577 70432081 70448669 70475813 70482637 70484461 70485469 70490569 70503593 70574173 70585693 70645409 70663657 70685837 70709909 71259701 71259893 71261317 71604517 71604529 71914933 72011449 72011809 72048113 72062257 72062489 72067181 72203473 72203513 72270113 72271217 72313321 72326113 72332233 72332257 72333889 72353213 72421493 87851969 87853069 89935973 89937677 89938561 89940953 89941589 89964913 90007633 90011021 90011773 90016897 90019733 90020257 90051109 90051173 90136957 202673453 202866649 202870021 202881389 202897241 202903321 203301173 203301457 203301797 203302381 206443357 206446441 206447893 206449157 206484793 206514577 206531777 206532901 206539037 206539121 206539153 206539373 206539601 206564537 206564681 206602681 206602897 206613877 206614081 206636933 206637133 206642801 206642881 206653009 206665937 207824657 207933617 207964193 207964333 207964513 209274629 209292329 209292437 209293061 209297017 209303509 209304457 209304497 209380229 209390729 209479993 209480189 209499677 209502677 211887857 211912229 211915213 211919237 211940189 212118769 212119549 212124733 212149849 212168933 212169613 212176841 212862173 212997553 213161369 213227897 213239153 390767609 390783053 390802837 390895961 402489449 402492089 402614441 402625793 402635689 402635741 402635777 402642677 402670817 402788173 402795361 402831197 402834869 403173161 403174249 403397653 403401937 403402729 415301401 415301629 415302637 415304941 426013633 426014293 426155537 430961129 430961417 430974949 430979557 430979761 435031369 435032077 435052957 435053261 435090749 435098393 435098501 435098753 435099593 435100933 435132277 436555673 436560977 436626709 436689257 437392957 437522401 437605409 437699849 437703457 437703797 437704781 437764441 437768033 437768257 437808881 437845297 437862037 437877929 451445021 451757717 451757821 451764361 453602309 453612541 453613357 453613649 453667813 453672221 509656897 510610193 510617209 510647861 510647953 510652481 511498313 511500877 511509841 511570597 511570733 512279249 512524093 512524889 512525653 512526017 512590217 512590361 512590373 512592989 512607157 513134021 513146189 513243281 513243641 513248629 513252449 513396893 513559577 513643381 513643573 513649393 515232449 515232833 515232877 515243717 515423497 515423521 515423977 515432017 515432069 515447341 515447473 515453881 515517053 515518777 515574593 515602589 515602973 515612221 515626481 515628901 515642837 515721593 515928137 515930137 515930521 515931061 515939297 663825733 663868973 663870569 663870917 664028941 664029001 664031897 664032001 664040549 664423313 669056321 669080453 669086557 670484557 670502341 670535329 670536137 670542797 670574101 670574813 670585313 670674481 670675157 670681097 671018009 671019593 671019949 671051321 671051357 671084669 671096501 671099909 671104673 671140397 671140469 671141941 671164717 671178817 671179477 671184809 671224033 671225249 671235209 671484277 671501141 671554193 671559269 671564809 671593337 671650961 671650997 671651993 671659229 671667709 672861617 672861949 675572477 675579353 675582877 675674213 682402621 682409177 682669621 682730669 683037629 683049893 683117629 683118673 683152193 683152409 683170613 683176253 683190397 683193529 683193949 683219221 684073009
The gaps between these numbers vary a lot. The record gaps are: n a(n) a(n)-a(n-1) fraction --- -------------- -------------- --------- 1 37 37 1.000000 5 3,821 3,708 0.980110 8 28,661 24,004 0.968180 17 133,981 76,680 0.779431 18 16,501,361 16,367,380 0.998209 19 46,178,761 29,677,400 0.999360 143 202,673,453 112,536,496 0.782962 214 390,767,609 177,528,456 0.860394 423 >4,295,135,754 >3,611,062,745 >0.919011
Here the first column is n, the second is a(n), the third column is the size of the gap between a(n-1) and a(n) and the last column is the fraction of a(n) accounted for by the sum of the preceding record gaps. The last line is a lower bound on the next gap, based on the state of affairs when my program stopped. So the path looks mostly like a very few very long meanders, separated by lots more (relatively) tiny excursions. (But note the 3 record gaps in a row ending at a(17), a(18) and a(19).) I wonder if the path ever returns after this point. I suspect that's a very deep question -- certainly beyond my ken.
On Tue, Dec 17, 2019 at 2:14 PM Neil Sloane <njasloane@gmail.com> wrote:
On Tue, Dec 17, 2019 at 4:24 PM Éric Angelini <eric.angelini@skynet.be> wrote:
Thank you Walter! This is an incentive for ski slopes! Best, É.
Le 17 déc. 2019 à 20:05, Walter Trump <w@trump.de> a écrit :
The next Boustrophedon primes are 3821 and 3989. Here is a continuation of Eric's arrangement of numbers (you cannot
read
the
digits, they are too small): https//www.trump.de/Boustrophedon-Primes.png
Walter
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Is there any reason to think this sequence is infinite? Or even any reason to think that the boustrophedon writing reaches the zero column infinitely often? (That might happen even if it places a *prime* in that column only finitely many times.) --Michael On Thu, Dec 19, 2019 at 3:25 AM Neil Sloane <njasloane@gmail.com> wrote:
The OEIS entry for Éric Angelini's Boustrophedon primes, A330339, has been greatly expanded, thanks to extensive computations by Hans Havermann (Tom Duff'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.
On Wed, Dec 18, 2019 at 7:55 PM Tom Duff <td@pixar.com> wrote:
I wrote a little program and computed the first 2^32 (4294967296) steps of the walk. Unless my program is busted, there are 422 boustrophedon primes up to 4294967296: 37 53 89 113 3821 3989 4657 28661 29021 41641 41669 44249 50909 56053 57041 57301 133981 16501361 46178761 47633441 47633477 47722049 47736121 47774621 47803477 47810209 47835013 47835341 47854969 47862413 47865017 49448573 49448617 49449133 49464241 50143469 50143601 53757637 53757797 53757881 56497153 56527117 56707681 56708429 56713697 56740009 56741309 56808877 56932357 56932801 56976697 57364081 57369617 57488653 57489233 57490997 57493861 57494573 57566141 57567161 57570521 57571009 57577061 57788597 57808649 57811097 57811609 68807029 68812013 68835721 68836129 68836333 68839289 68847853 69985609 69985633 69987377 69996869 69999961 70001873 70012781 70015217 70095209 70095661 70113833 70114181 70144541 70420577 70432081 70448669 70475813 70482637 70484461 70485469 70490569 70503593 70574173 70585693 70645409 70663657 70685837 70709909 71259701 71259893 71261317 71604517 71604529 71914933 72011449 72011809 72048113 72062257 72062489 72067181 72203473 72203513 72270113 72271217 72313321 72326113 72332233 72332257 72333889 72353213 72421493 87851969 87853069 89935973 89937677 89938561 89940953 89941589 89964913 90007633 90011021 90011773 90016897 90019733 90020257 90051109 90051173 90136957 202673453 202866649 202870021 202881389 202897241 202903321 203301173 203301457 203301797 203302381 206443357 206446441 206447893 206449157 206484793 206514577 206531777 206532901 206539037 206539121 206539153 206539373 206539601 206564537 206564681 206602681 206602897 206613877 206614081 206636933 206637133 206642801 206642881 206653009 206665937 207824657 207933617 207964193 207964333 207964513 209274629 209292329 209292437 209293061 209297017 209303509 209304457 209304497 209380229 209390729 209479993 209480189 209499677 209502677 211887857 211912229 211915213 211919237 211940189 212118769 212119549 212124733 212149849 212168933 212169613 212176841 212862173 212997553 213161369 213227897 213239153 390767609 390783053 390802837 390895961 402489449 402492089 402614441 402625793 402635689 402635741 402635777 402642677 402670817 402788173 402795361 402831197 402834869 403173161 403174249 403397653 403401937 403402729 415301401 415301629 415302637 415304941 426013633 426014293 426155537 430961129 430961417 430974949 430979557 430979761 435031369 435032077 435052957 435053261 435090749 435098393 435098501 435098753 435099593 435100933 435132277 436555673 436560977 436626709 436689257 437392957 437522401 437605409 437699849 437703457 437703797 437704781 437764441 437768033 437768257 437808881 437845297 437862037 437877929 451445021 451757717 451757821 451764361 453602309 453612541 453613357 453613649 453667813 453672221 509656897 510610193 510617209 510647861 510647953 510652481 511498313 511500877 511509841 511570597 511570733 512279249 512524093 512524889 512525653 512526017 512590217 512590361 512590373 512592989 512607157 513134021 513146189 513243281 513243641 513248629 513252449 513396893 513559577 513643381 513643573 513649393 515232449 515232833 515232877 515243717 515423497 515423521 515423977 515432017 515432069 515447341 515447473 515453881 515517053 515518777 515574593 515602589 515602973 515612221 515626481 515628901 515642837 515721593 515928137 515930137 515930521 515931061 515939297 663825733 663868973 663870569 663870917 664028941 664029001 664031897 664032001 664040549 664423313 669056321 669080453 669086557 670484557 670502341 670535329 670536137 670542797 670574101 670574813 670585313 670674481 670675157 670681097 671018009 671019593 671019949 671051321 671051357 671084669 671096501 671099909 671104673 671140397 671140469 671141941 671164717 671178817 671179477 671184809 671224033 671225249 671235209 671484277 671501141 671554193 671559269 671564809 671593337 671650961 671650997 671651993 671659229 671667709 672861617 672861949 675572477 675579353 675582877 675674213 682402621 682409177 682669621 682730669 683037629 683049893 683117629 683118673 683152193 683152409 683170613 683176253 683190397 683193529 683193949 683219221 684073009
The gaps between these numbers vary a lot. The record gaps are: n a(n) a(n)-a(n-1) fraction --- -------------- -------------- --------- 1 37 37 1.000000 5 3,821 3,708 0.980110 8 28,661 24,004 0.968180 17 133,981 76,680 0.779431 18 16,501,361 16,367,380 0.998209 19 46,178,761 29,677,400 0.999360 143 202,673,453 112,536,496 0.782962 214 390,767,609 177,528,456 0.860394 423 >4,295,135,754 >3,611,062,745 >0.919011
Here the first column is n, the second is a(n), the third column is the size of the gap between a(n-1) and a(n) and the last column is the fraction of a(n) accounted for by the sum of the preceding record gaps. The last line is a lower bound on the next gap, based on the state of affairs when my program stopped. So the path looks mostly like a very few very long meanders, separated by lots more (relatively) tiny excursions. (But note the 3 record gaps in a row ending at a(17), a(18) and a(19).) I wonder if the path ever returns after this point. I suspect that's a very deep question -- certainly beyond my ken.
On Tue, Dec 17, 2019 at 2:14 PM Neil Sloane <njasloane@gmail.com> wrote:
On Tue, Dec 17, 2019 at 4:24 PM Éric Angelini <eric.angelini@skynet.be
wrote:
Thank you Walter! This is an incentive for ski slopes! Best, É.
Le 17 déc. 2019 à 20:05, Walter Trump <w@trump.de> a écrit :
The next Boustrophedon primes are 3821 and 3989. Here is a continuation of Eric's arrangement of numbers (you cannot read the digits, they are too small): https//www.trump.de/Boustrophedon-Primes.png
Walter
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-- Forewarned is worth an octopus in the bush.
My guess is the Bous primes are infinite in number. The b-file for A330339 gives 10000 terms, so there is no shortage. Hans's second graph in A330545 suggests that there are times when the path crosses the x-axis and dithers around there for a while, picking up a lot of hits right on the x-axis, something that is borne out by the b-file. And it seems clear this is a coin-tossing sequence, meaning that there will be infinitely many sign changes, separated by runs that stay on one side or the other of the x-axis for a long time. On Fri, Dec 20, 2019 at 11:30 AM Michael Kleber <michael.kleber@gmail.com> wrote:
Is there any reason to think this sequence is infinite?
Or even any reason to think that the boustrophedon writing reaches the zero column infinitely often? (That might happen even if it places a *prime* in that column only finitely many times.)
--Michael
On Thu, Dec 19, 2019 at 3:25 AM Neil Sloane <njasloane@gmail.com> wrote:
The OEIS entry for Éric Angelini's Boustrophedon primes, A330339, has been greatly expanded, thanks to extensive computations by Hans Havermann (Tom Duff'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.
On Wed, Dec 18, 2019 at 7:55 PM Tom Duff <td@pixar.com> wrote:
I wrote a little program and computed the first 2^32 (4294967296) steps of the walk. Unless my program is busted, there are 422 boustrophedon primes up to 4294967296: 37 53 89 113 3821 3989 4657 28661 29021 41641 41669 44249 50909 56053 57041 57301 133981 16501361 46178761 47633441 47633477 47722049 47736121 47774621 47803477 47810209 47835013 47835341 47854969 47862413 47865017 49448573 49448617 49449133 49464241 50143469 50143601 53757637 53757797 53757881 56497153 56527117 56707681 56708429 56713697 56740009 56741309 56808877 56932357 56932801 56976697 57364081 57369617 57488653 57489233 57490997 57493861 57494573 57566141 57567161 57570521 57571009 57577061 57788597 57808649 57811097 57811609 68807029 68812013 68835721 68836129 68836333 68839289 68847853 69985609 69985633 69987377 69996869 69999961 70001873 70012781 70015217 70095209 70095661 70113833 70114181 70144541 70420577 70432081 70448669 70475813 70482637 70484461 70485469 70490569 70503593 70574173 70585693 70645409 70663657 70685837 70709909 71259701 71259893 71261317 71604517 71604529 71914933 72011449 72011809 72048113 72062257 72062489 72067181 72203473 72203513 72270113 72271217 72313321 72326113 72332233 72332257 72333889 72353213 72421493 87851969 87853069 89935973 89937677 89938561 89940953 89941589 89964913 90007633 90011021 90011773 90016897 90019733 90020257 90051109 90051173 90136957 202673453 202866649 202870021 202881389 202897241 202903321 203301173 203301457 203301797 203302381 206443357 206446441 206447893 206449157 206484793 206514577 206531777 206532901 206539037 206539121 206539153 206539373 206539601 206564537 206564681 206602681 206602897 206613877 206614081 206636933 206637133 206642801 206642881 206653009 206665937 207824657 207933617 207964193 207964333 207964513 209274629 209292329 209292437 209293061 209297017 209303509 209304457 209304497 209380229 209390729 209479993 209480189 209499677 209502677 211887857 211912229 211915213 211919237 211940189 212118769 212119549 212124733 212149849 212168933 212169613 212176841 212862173 212997553 213161369 213227897 213239153 390767609 390783053 390802837 390895961 402489449 402492089 402614441 402625793 402635689 402635741 402635777 402642677 402670817 402788173 402795361 402831197 402834869 403173161 403174249 403397653 403401937 403402729 415301401 415301629 415302637 415304941 426013633 426014293 426155537 430961129 430961417 430974949 430979557 430979761 435031369 435032077 435052957 435053261 435090749 435098393 435098501 435098753 435099593 435100933 435132277 436555673 436560977 436626709 436689257 437392957 437522401 437605409 437699849 437703457 437703797 437704781 437764441 437768033 437768257 437808881 437845297 437862037 437877929 451445021 451757717 451757821 451764361 453602309 453612541 453613357 453613649 453667813 453672221 509656897 510610193 510617209 510647861 510647953 510652481 511498313 511500877 511509841 511570597 511570733 512279249 512524093 512524889 512525653 512526017 512590217 512590361 512590373 512592989 512607157 513134021 513146189 513243281 513243641 513248629 513252449 513396893 513559577 513643381 513643573 513649393 515232449 515232833 515232877 515243717 515423497 515423521 515423977 515432017 515432069 515447341 515447473 515453881 515517053 515518777 515574593 515602589 515602973 515612221 515626481 515628901 515642837 515721593 515928137 515930137 515930521 515931061 515939297 663825733 663868973 663870569 663870917 664028941 664029001 664031897 664032001 664040549 664423313 669056321 669080453 669086557 670484557 670502341 670535329 670536137 670542797 670574101 670574813 670585313 670674481 670675157 670681097 671018009 671019593 671019949 671051321 671051357 671084669 671096501 671099909 671104673 671140397 671140469 671141941 671164717 671178817 671179477 671184809 671224033 671225249 671235209 671484277 671501141 671554193 671559269 671564809 671593337 671650961 671650997 671651993 671659229 671667709 672861617 672861949 675572477 675579353 675582877 675674213 682402621 682409177 682669621 682730669 683037629 683049893 683117629 683118673 683152193 683152409 683170613 683176253 683190397 683193529 683193949 683219221 684073009
The gaps between these numbers vary a lot. The record gaps are: n a(n) a(n)-a(n-1) fraction --- -------------- -------------- --------- 1 37 37 1.000000 5 3,821 3,708 0.980110 8 28,661 24,004 0.968180 17 133,981 76,680 0.779431 18 16,501,361 16,367,380 0.998209 19 46,178,761 29,677,400 0.999360 143 202,673,453 112,536,496 0.782962 214 390,767,609 177,528,456 0.860394 423 >4,295,135,754 >3,611,062,745 >0.919011
Here the first column is n, the second is a(n), the third column is the size of the gap between a(n-1) and a(n) and the last column is the fraction of a(n) accounted for by the sum of the preceding record gaps. The last line is a lower bound on the next gap, based on the state of affairs when my program stopped. So the path looks mostly like a very few very long meanders, separated by lots more (relatively) tiny excursions. (But note the 3 record gaps in a row ending at a(17), a(18) and a(19).) I wonder if the path ever returns after this point. I suspect that's a very deep question -- certainly beyond my ken.
On Tue, Dec 17, 2019 at 2:14 PM Neil Sloane <njasloane@gmail.com> wrote:
On Tue, Dec 17, 2019 at 4:24 PM Éric Angelini <
eric.angelini@skynet.be
wrote:
Thank you Walter! This is an incentive for ski slopes! Best, É.
Le 17 déc. 2019 à 20:05, Walter Trump <w@trump.de> a écrit :
The next Boustrophedon primes are 3821 and 3989. Here is a continuation of Eric's arrangement of numbers (you cannot read the digits, they are too small): https//www.trump.de/Boustrophedon-Primes.png
Walter
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-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
The race between primes that are 3-mod-4 and 1-mod-4 has its lead change infinitely often, according to a result of Littlewood from 1914 (see more at https://en.wikipedia.org/wiki/Prime_number_theorem#Prime_number_race and Granville & Martin's lovely Monthly paper https://www.jstor.org/stable/27641834?seq=1). I don't think there's any direct way that answers the present question, but it seems similar enough in nature that I wonder if any of the generalized prime races work is relevant. The boustrophedon situation cares about the lengths of gaps between consecutive primes, though, so maybe it's hopelessly different. --Michael On Fri, Dec 20, 2019 at 12:40 PM Neil Sloane <njasloane@gmail.com> wrote:
My guess is the Bous primes are infinite in number. The b-file for A330339 gives 10000 terms, so there is no shortage. Hans's second graph in A330545 suggests that there are times when the path crosses the x-axis and dithers around there for a while, picking up a lot of hits right on the x-axis, something that is borne out by the b-file.
And it seems clear this is a coin-tossing sequence, meaning that there will be infinitely many sign changes, separated by runs that stay on one side or the other of the x-axis for a long time.
On Fri, Dec 20, 2019 at 11:30 AM Michael Kleber <michael.kleber@gmail.com> wrote:
Is there any reason to think this sequence is infinite?
Or even any reason to think that the boustrophedon writing reaches the zero column infinitely often? (That might happen even if it places a *prime* in that column only finitely many times.)
--Michael
On Thu, Dec 19, 2019 at 3:25 AM Neil Sloane <njasloane@gmail.com> wrote:
The OEIS entry for Éric Angelini's Boustrophedon primes, A330339, has been greatly expanded, thanks to extensive computations by Hans Havermann (Tom Duff'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.
On Wed, Dec 18, 2019 at 7:55 PM Tom Duff <td@pixar.com> wrote:
I wrote a little program and computed the first 2^32 (4294967296) steps of the walk. Unless my program is busted, there are 422 boustrophedon primes up to 4294967296: 37 53 89 113 3821 3989 4657 28661 29021 41641 41669 44249 50909 56053 57041 57301 133981 16501361 46178761 47633441 47633477 47722049 47736121 47774621 47803477 47810209 47835013 47835341 47854969 47862413 47865017 49448573 49448617 49449133 49464241 50143469 50143601 53757637 53757797 53757881 56497153 56527117 56707681 56708429 56713697 56740009 56741309 56808877 56932357 56932801 56976697 57364081 57369617 57488653 57489233 57490997 57493861 57494573 57566141 57567161 57570521 57571009 57577061 57788597 57808649 57811097 57811609 68807029 68812013 68835721 68836129 68836333 68839289 68847853 69985609 69985633 69987377 69996869 69999961 70001873 70012781 70015217 70095209 70095661 70113833 70114181 70144541 70420577 70432081 70448669 70475813 70482637 70484461 70485469 70490569 70503593 70574173 70585693 70645409 70663657 70685837 70709909 71259701 71259893 71261317 71604517 71604529 71914933 72011449 72011809 72048113 72062257 72062489 72067181 72203473 72203513 72270113 72271217 72313321 72326113 72332233 72332257 72333889 72353213 72421493 87851969 87853069 89935973 89937677 89938561 89940953 89941589 89964913 90007633 90011021 90011773 90016897 90019733 90020257 90051109 90051173 90136957 202673453 202866649 202870021 202881389 202897241 202903321 203301173 203301457 203301797 203302381 206443357 206446441 206447893 206449157 206484793 206514577 206531777 206532901 206539037 206539121 206539153 206539373 206539601 206564537 206564681 206602681 206602897 206613877 206614081 206636933 206637133 206642801 206642881 206653009 206665937 207824657 207933617 207964193 207964333 207964513 209274629 209292329 209292437 209293061 209297017 209303509 209304457 209304497 209380229 209390729 209479993 209480189 209499677 209502677 211887857 211912229 211915213 211919237 211940189 212118769 212119549 212124733 212149849 212168933 212169613 212176841 212862173 212997553 213161369 213227897 213239153 390767609 390783053 390802837 390895961 402489449 402492089 402614441 402625793 402635689 402635741 402635777 402642677 402670817 402788173 402795361 402831197 402834869 403173161 403174249 403397653 403401937 403402729 415301401 415301629 415302637 415304941 426013633 426014293 426155537 430961129 430961417 430974949 430979557 430979761 435031369 435032077 435052957 435053261 435090749 435098393 435098501 435098753 435099593 435100933 435132277 436555673 436560977 436626709 436689257 437392957 437522401 437605409 437699849 437703457 437703797 437704781 437764441 437768033 437768257 437808881 437845297 437862037 437877929 451445021 451757717 451757821 451764361 453602309 453612541 453613357 453613649 453667813 453672221 509656897 510610193 510617209 510647861 510647953 510652481 511498313 511500877 511509841 511570597 511570733 512279249 512524093 512524889 512525653 512526017 512590217 512590361 512590373 512592989 512607157 513134021 513146189 513243281 513243641 513248629 513252449 513396893 513559577 513643381 513643573 513649393 515232449 515232833 515232877 515243717 515423497 515423521 515423977 515432017 515432069 515447341 515447473 515453881 515517053 515518777 515574593 515602589 515602973 515612221 515626481 515628901 515642837 515721593 515928137 515930137 515930521 515931061 515939297 663825733 663868973 663870569 663870917 664028941 664029001 664031897 664032001 664040549 664423313 669056321 669080453 669086557 670484557 670502341 670535329 670536137 670542797 670574101 670574813 670585313 670674481 670675157 670681097 671018009 671019593 671019949 671051321 671051357 671084669 671096501 671099909 671104673 671140397 671140469 671141941 671164717 671178817 671179477 671184809 671224033 671225249 671235209 671484277 671501141 671554193 671559269 671564809 671593337 671650961 671650997 671651993 671659229 671667709 672861617 672861949 675572477 675579353 675582877 675674213 682402621 682409177 682669621 682730669 683037629 683049893 683117629 683118673 683152193 683152409 683170613 683176253 683190397 683193529 683193949 683219221 684073009
The gaps between these numbers vary a lot. The record gaps are: n a(n) a(n)-a(n-1) fraction --- -------------- -------------- --------- 1 37 37 1.000000 5 3,821 3,708 0.980110 8 28,661 24,004 0.968180 17 133,981 76,680 0.779431 18 16,501,361 16,367,380 0.998209 19 46,178,761 29,677,400 0.999360 143 202,673,453 112,536,496 0.782962 214 390,767,609 177,528,456 0.860394 423 >4,295,135,754 >3,611,062,745 >0.919011
Here the first column is n, the second is a(n), the third column is the size of the gap between a(n-1) and a(n) and the last column is the fraction of a(n) accounted for by the sum of the preceding record gaps. The last line is a lower bound on the next gap, based on the state of affairs when my program stopped. So the path looks mostly like a very few very long meanders, separated by lots more (relatively) tiny excursions. (But note the 3 record gaps in a row ending at a(17), a(18) and a(19).) I wonder if the path ever returns after this point. I suspect that's a very deep question -- certainly beyond my ken.
On Tue, Dec 17, 2019 at 2:14 PM Neil Sloane <njasloane@gmail.com> wrote:
On Tue, Dec 17, 2019 at 4:24 PM Éric Angelini <
eric.angelini@skynet.be
wrote:
Thank you Walter! This is an incentive for ski slopes! Best, É.
> Le 17 déc. 2019 à 20:05, Walter Trump <w@trump.de> a écrit : > > The next Boustrophedon primes are 3821 and 3989. > Here is a continuation of Eric's arrangement of numbers (you cannot read the > digits, they are too small): > https//www.trump.de/Boustrophedon-Primes.png > > Walter > > > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Hello Math-Fun, Let's call S the lexicographically earliest sequence of distinct integers > 0 with the following property: «The a(n)th digit of S is the number of digits used by a(n)». [for example, if the 85th digit of S is a 4, then a(85) uses exactly 4 digits] Questions: 1) how many terms has S? 2) what is the last term of S? There are enough elements above to ask, but if you want to see how S starts, and have some more explanations, please click here (link to my personal blog): https://bit.ly/2Ss2BFE Best, É.
The property was wrong, soooooooooorry! Now ok... Hello Math-Fun, Let's call S the lexicographically earliest sequence of distinct integers > 0 with the following property: > «The nth digit of S is the number of digits used by a(n)». [for example, if the 85th digit of S is a 4, then a(85) uses exactly 4 digits] Questions: 1) how many terms has S? 2) what is the last term of S? There are enough elements above to answer, but if you want to see how S starts, and have some more explanations, please click here (link to my personal blog): https://bit.ly/2Ss2BFE Best, É.
participants (7)
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Allan Wechsler -
Michael Kleber -
Neil Sloane -
Tom Duff -
Walter Trump -
Éric Angelini -
Éric Angelini