[math-fun] What the Eck?
Neil Sloane has Numberphiled a perplexing integer sequence: https://www.youtube.com/watch?v=etMJxB-igrc —rwg
Ouch... this one looks like it could lead to some serious brain damage. If A171868 lists all integers, then so too does A181391. Due to the rule, A171868 has much slower growth and, as far as I can tell, could even be bounded above. Does A171868 always continue to grow? I don't know. --Brad On Tue, Jun 11, 2019 at 2:48 AM Bill Gosper <billgosper@gmail.com> wrote:
Neil Sloane has Numberphiled a perplexing integer sequence: https://www.youtube.com/watch?v=etMJxB-igrc —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Following along Brad's comments, here are the first occurrences of pairs starting with 0 in Van Eck's sequence. Anyone want to try to fill in the value for (0,32) (or (0,n) for n>34)? Pair 0 0 at 1 Pair 0 1 at 2 Pair 0 2 at 4 Pair 0 3 at 21 Pair 0 4 at 25 Pair 0 5 at 11 Pair 0 6 at 31 Pair 0 7 at 277 Pair 0 8 at 389 Pair 0 9 at 82 Pair 0 10 at 226 Pair 0 11 at 727 Pair 0 12 at 2,936 Pair 0 13 at 1,409 Pair 0 14 at 7,719 Pair 0 15 at 5,625 Pair 0 16 at 5,681 Pair 0 17 at 85,999 Pair 0 18 at 26,707 Pair 0 19 at 546,291 Pair 0 20 at 1,112,930 Pair 0 21 at 702,576 Pair 0 22 at 3,425,418 Pair 0 23 at 10,537,361 Pair 0 24 at 21,301,907 Pair 0 25 at 217,230,901 Pair 0 26 at 108,698,092 Pair 0 27 at 32,381,775 Pair 0 28 at 846,522,987 Pair 0 29 at 851,764,847 Pair 0 30 at 11,692,311,326 Pair 0 31 at 46,163,898,988 Pair 0 32 at ??? Pair 0 33 at 118,456,929,919 Pair 0 34 at 250,327,022,558 Pair 0 35 at ??? On Wed, Jun 12, 2019 at 12:25 AM Brad Klee <bradklee@gmail.com> wrote:
Ouch... this one looks like it could lead to some serious brain damage.
If A171868 lists all integers, then so too does A181391. Due to the rule, A171868 has much slower growth and, as far as I can tell, could even be bounded above. Does A171868 always continue to grow?
I don't know.
--Brad
On Tue, Jun 11, 2019 at 2:48 AM Bill Gosper <billgosper@gmail.com> wrote:
Neil Sloane has Numberphiled a perplexing integer sequence: https://www.youtube.com/watch?v=etMJxB-igrc —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Also, at this point I've shown if there's an integer missing from Van Eck's sequence, it is at least 519,068,589. And 6's are increasingly rare, more so than other small integers (up to about 250B entries). Initially 3's are rare, but this characteristic moves to 4's, then 5's, and now 6's. One could expect this to continue . . . On Sun, Jun 16, 2019 at 10:39 AM Tomas Rokicki <rokicki@gmail.com> wrote:
Following along Brad's comments, here are the first occurrences of pairs starting with 0 in Van Eck's sequence. Anyone want to try to fill in the value for (0,32) (or (0,n) for n>34)?
Pair 0 0 at 1 Pair 0 1 at 2 Pair 0 2 at 4 Pair 0 3 at 21 Pair 0 4 at 25 Pair 0 5 at 11 Pair 0 6 at 31 Pair 0 7 at 277 Pair 0 8 at 389 Pair 0 9 at 82 Pair 0 10 at 226 Pair 0 11 at 727 Pair 0 12 at 2,936 Pair 0 13 at 1,409 Pair 0 14 at 7,719 Pair 0 15 at 5,625 Pair 0 16 at 5,681 Pair 0 17 at 85,999 Pair 0 18 at 26,707 Pair 0 19 at 546,291 Pair 0 20 at 1,112,930 Pair 0 21 at 702,576 Pair 0 22 at 3,425,418 Pair 0 23 at 10,537,361 Pair 0 24 at 21,301,907 Pair 0 25 at 217,230,901 Pair 0 26 at 108,698,092 Pair 0 27 at 32,381,775 Pair 0 28 at 846,522,987 Pair 0 29 at 851,764,847 Pair 0 30 at 11,692,311,326 Pair 0 31 at 46,163,898,988 Pair 0 32 at ??? Pair 0 33 at 118,456,929,919 Pair 0 34 at 250,327,022,558 Pair 0 35 at ???
On Wed, Jun 12, 2019 at 12:25 AM Brad Klee <bradklee@gmail.com> wrote:
Ouch... this one looks like it could lead to some serious brain damage.
If A171868 lists all integers, then so too does A181391. Due to the rule, A171868 has much slower growth and, as far as I can tell, could even be bounded above. Does A171868 always continue to grow?
I don't know.
--Brad
On Tue, Jun 11, 2019 at 2:48 AM Bill Gosper <billgosper@gmail.com> wrote:
Neil Sloane has Numberphiled a perplexing integer sequence: https://www.youtube.com/watch?v=etMJxB-igrc —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Tomas, You might add this remark as a comment in A181391: "if there's an integer missing [from Van Eck's sequence], it is at least 519,068,589." Because other people are surely looking at the same question, so it is important to keep a record of how far the search has gone. (But remember in the OEIS numbers don't have internal commas or spaces!) Brad and Tomas, This sequence should be added to the OEIS, if you have not already done so! (with offset 0): 1,2,4,21,25,11,... based on Pair 0 0 at 1 Pair 0 1 at 2 Pair 0 2 at 4 Pair 0 3 at 21 Pair 0 4 at 25 Pair 0 5 at 11 Pair 0 6 at 31 Pair 0 7 at 277 Interesting sequence. 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 Sun, Jun 16, 2019 at 1:46 PM Tomas Rokicki <rokicki@gmail.com> wrote:
Also, at this point I've shown if there's an integer missing from Van Eck's sequence, it is at least 519,068,589.
And 6's are increasingly rare, more so than other small integers (up to about 250B entries). Initially 3's are rare, but this characteristic moves to 4's, then 5's, and now 6's. One could expect this to continue . . .
On Sun, Jun 16, 2019 at 10:39 AM Tomas Rokicki <rokicki@gmail.com> wrote:
Following along Brad's comments, here are the first occurrences of pairs starting with 0 in Van Eck's sequence. Anyone want to try to fill in the value for (0,32) (or (0,n) for n>34)?
Pair 0 0 at 1 Pair 0 1 at 2 Pair 0 2 at 4 Pair 0 3 at 21 Pair 0 4 at 25 Pair 0 5 at 11 Pair 0 6 at 31 Pair 0 7 at 277 Pair 0 8 at 389 Pair 0 9 at 82 Pair 0 10 at 226 Pair 0 11 at 727 Pair 0 12 at 2,936 Pair 0 13 at 1,409 Pair 0 14 at 7,719 Pair 0 15 at 5,625 Pair 0 16 at 5,681 Pair 0 17 at 85,999 Pair 0 18 at 26,707 Pair 0 19 at 546,291 Pair 0 20 at 1,112,930 Pair 0 21 at 702,576 Pair 0 22 at 3,425,418 Pair 0 23 at 10,537,361 Pair 0 24 at 21,301,907 Pair 0 25 at 217,230,901 Pair 0 26 at 108,698,092 Pair 0 27 at 32,381,775 Pair 0 28 at 846,522,987 Pair 0 29 at 851,764,847 Pair 0 30 at 11,692,311,326 Pair 0 31 at 46,163,898,988 Pair 0 32 at ??? Pair 0 33 at 118,456,929,919 Pair 0 34 at 250,327,022,558 Pair 0 35 at ???
On Wed, Jun 12, 2019 at 12:25 AM Brad Klee <bradklee@gmail.com> wrote:
Ouch... this one looks like it could lead to some serious brain damage.
If A171868 lists all integers, then so too does A181391. Due to the rule, A171868 has much slower growth and, as far as I can tell, could even be bounded above. Does A171868 always continue to grow?
I don't know.
--Brad
On Tue, Jun 11, 2019 at 2:48 AM Bill Gosper <billgosper@gmail.com> wrote:
Neil Sloane has Numberphiled a perplexing integer sequence: https://www.youtube.com/watch?v=etMJxB-igrc —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Is anything known about the occurrences of consecutive pairs of the same number in the series? The pair 0,0 obviously occurs once, at the very beginning. And the pair 1,1 cannot occur, since it would produce an infinite repeating pattern. Other pairs do occur though (I believe 2, 3, 8, etc.) Tom
So far I see all pairs (a,a) occurring [except (1,1)] reasonably early (I've checked through a=1023). The longest one of those took to appear was 992, which occurred at 662,150,300. The pairs that seem to be taking the longest to appear are (0,a); after this, they are values (a,b) where either a or b is in the range 4..10, or where b-a is 5 or 6. The main thing I'm seeing is that the sequence is a series of ramps; a ramp starts with 0 and ends with a previously unseen value. Between these two values is a series of usually increasing numbers that appear to increase quickly. The length of the ramps varies but the average ramp length increases slowly. Most of the small integers that recur are the ramp length. If you look at the frequency of occurrence of values, you see a graph with some interesting and significant undulations. As the sequence continues those undulations move to the right, but the overall shape of the undulations appears pretty self-similar. I'll try to get some graphs up soon. Some late-appearing pairs where (a,b) are both less than 1024 are: See 0 34 at 250327022558 See 943 4 at 250470176885 See 9 937 at 250505969939 See 9 949 at 251219693325 See 924 4 at 251411817200 See 9 962 at 252043560162 See 10 708 at 252079357435 See 9 950 at 252247778126 See 577 583 at 252730372074 See 1 875 at 253071570424 See 481 5 at 254277582179 See 751 756 at 254389321902 See 841 4 at 254643579836 Frequences of occurrences; see the undulations? 0 23715597276 1 3858369548 2 1292639191 3 89031007 4 8625117 5 2302753 6 4589090 7 155877539 8 1847170967 9 6193152432 10 7958563024 11 6225475874 12 4061764482 13 2196264487 14 981661792 15 420357422 16 176297472 17 196811670 18 185357188 19 753266367 20 767663426 21 1032462039 22 870293065 23 485252403 24 382523736 25 120254552 26 109793239 27 145701135 28 167362679 29 317189883 30 619791368 31 648210091 32 672215340 33 577231661 34 462054056 35 283492644 36 173878733 37 146474139 38 165726872 39 234593490 40 318784422 41 408898611 42 448076370 On Sun, Jun 16, 2019 at 12:31 PM Tom Karzes <karzes@sonic.net> wrote:
Is anything known about the occurrences of consecutive pairs of the same number in the series? The pair 0,0 obviously occurs once, at the very beginning. And the pair 1,1 cannot occur, since it would produce an infinite repeating pattern. Other pairs do occur though (I believe 2, 3, 8, etc.)
Tom
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interesting! so the distribution is nonuniform, even among small numbers? what’s a good numerical criterion to see if this persists? do we believe that asymptotically the distribution of numbers becomes stable? C
On Jun 16, 2019, at 11:46 AM, Tomas Rokicki <rokicki@gmail.com> wrote:
Also, at this point I've shown if there's an integer missing from Van Eck's sequence, it is at least 519,068,589.
And 6's are increasingly rare, more so than other small integers (up to about 250B entries). Initially 3's are rare, but this characteristic moves to 4's, then 5's, and now 6's. One could expect this to continue . . .
On Sun, Jun 16, 2019 at 10:39 AM Tomas Rokicki <rokicki@gmail.com> wrote:
Following along Brad's comments, here are the first occurrences of pairs starting with 0 in Van Eck's sequence. Anyone want to try to fill in the value for (0,32) (or (0,n) for n>34)?
Pair 0 0 at 1 Pair 0 1 at 2 Pair 0 2 at 4 Pair 0 3 at 21 Pair 0 4 at 25 Pair 0 5 at 11 Pair 0 6 at 31 Pair 0 7 at 277 Pair 0 8 at 389 Pair 0 9 at 82 Pair 0 10 at 226 Pair 0 11 at 727 Pair 0 12 at 2,936 Pair 0 13 at 1,409 Pair 0 14 at 7,719 Pair 0 15 at 5,625 Pair 0 16 at 5,681 Pair 0 17 at 85,999 Pair 0 18 at 26,707 Pair 0 19 at 546,291 Pair 0 20 at 1,112,930 Pair 0 21 at 702,576 Pair 0 22 at 3,425,418 Pair 0 23 at 10,537,361 Pair 0 24 at 21,301,907 Pair 0 25 at 217,230,901 Pair 0 26 at 108,698,092 Pair 0 27 at 32,381,775 Pair 0 28 at 846,522,987 Pair 0 29 at 851,764,847 Pair 0 30 at 11,692,311,326 Pair 0 31 at 46,163,898,988 Pair 0 32 at ??? Pair 0 33 at 118,456,929,919 Pair 0 34 at 250,327,022,558 Pair 0 35 at ???
On Wed, Jun 12, 2019 at 12:25 AM Brad Klee <bradklee@gmail.com> wrote:
Ouch... this one looks like it could lead to some serious brain damage.
If A171868 lists all integers, then so too does A181391. Due to the rule, A171868 has much slower growth and, as far as I can tell, could even be bounded above. Does A171868 always continue to grow?
I don't know.
--Brad
On Tue, Jun 11, 2019 at 2:48 AM Bill Gosper <billgosper@gmail.com> wrote:
Neil Sloane has Numberphiled a perplexing integer sequence: https://www.youtube.com/watch?v=etMJxB-igrc —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I don't. I see this distribution of a sum of a series of Poisson-like distributions. In the ramps, the first number is always 0. The second number follows the distribution of the length of the ramps, and is Poisson-like (and small). The third number in the ramp follows (very roughly) the distribution of (0,n) pairs, where n is the second number in the ramp. And so on and so forth. The distributions are sufficiently disjoint where the second order effects stay somewhat small, and the distributions are clearly separable visually. The means of the distributions appear to increase roughly exponentially. There are second-order effects, but I think this is the dominant effect and explains the undulations in the overall distribution. It would be interesting to model this as, or fit this to, a random process with much less state. This won't give particular discrete results but it may give some insight into the behavior. Towards this end it may be useful to mark the peaks of the distributions and how those peaks migrate positivewards. -tom On Sun, Jun 16, 2019 at 1:18 PM Cris Moore <moore@santafe.edu> wrote:
interesting! so the distribution is nonuniform, even among small numbers? what’s a good numerical criterion to see if this persists? do we believe that asymptotically the distribution of numbers becomes stable?
C
On Jun 16, 2019, at 11:46 AM, Tomas Rokicki <rokicki@gmail.com> wrote:
Also, at this point I've shown if there's an integer missing from Van Eck's sequence, it is at least 519,068,589.
And 6's are increasingly rare, more so than other small integers (up to about 250B entries). Initially 3's are rare, but this characteristic moves to 4's, then 5's, and now 6's. One could expect this to continue . . .
On Sun, Jun 16, 2019 at 10:39 AM Tomas Rokicki <rokicki@gmail.com> wrote:
Following along Brad's comments, here are the first occurrences of pairs starting with 0 in Van Eck's sequence. Anyone want to try to fill in the value for (0,32) (or (0,n) for n>34)?
Pair 0 0 at 1 Pair 0 1 at 2 Pair 0 2 at 4 Pair 0 3 at 21 Pair 0 4 at 25 Pair 0 5 at 11 Pair 0 6 at 31 Pair 0 7 at 277 Pair 0 8 at 389 Pair 0 9 at 82 Pair 0 10 at 226 Pair 0 11 at 727 Pair 0 12 at 2,936 Pair 0 13 at 1,409 Pair 0 14 at 7,719 Pair 0 15 at 5,625 Pair 0 16 at 5,681 Pair 0 17 at 85,999 Pair 0 18 at 26,707 Pair 0 19 at 546,291 Pair 0 20 at 1,112,930 Pair 0 21 at 702,576 Pair 0 22 at 3,425,418 Pair 0 23 at 10,537,361 Pair 0 24 at 21,301,907 Pair 0 25 at 217,230,901 Pair 0 26 at 108,698,092 Pair 0 27 at 32,381,775 Pair 0 28 at 846,522,987 Pair 0 29 at 851,764,847 Pair 0 30 at 11,692,311,326 Pair 0 31 at 46,163,898,988 Pair 0 32 at ??? Pair 0 33 at 118,456,929,919 Pair 0 34 at 250,327,022,558 Pair 0 35 at ???
On Wed, Jun 12, 2019 at 12:25 AM Brad Klee <bradklee@gmail.com> wrote:
Ouch... this one looks like it could lead to some serious brain damage.
If A171868 lists all integers, then so too does A181391. Due to the rule, A171868 has much slower growth and, as far as I can tell, could even be bounded above. Does A171868 always continue to grow?
I don't know.
--Brad
On Tue, Jun 11, 2019 at 2:48 AM Bill Gosper <billgosper@gmail.com> wrote:
Neil Sloane has Numberphiled a perplexing integer sequence: https://www.youtube.com/watch?v=etMJxB-igrc —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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participants (6)
-
Bill Gosper -
Brad Klee -
Cris Moore -
Neil Sloane -
Tom Karzes -
Tomas Rokicki